Conic Section Notes for IIT JEE contains important formulas, identities, terms, properties, theorem, etc which are to be remembered to easily solve problems in the IIT JEE exam. Revising these IIT JEE notes of Conic Section will help students to quickly go through the main points of Conic Sections. We provide complete notes of Conic Section for IIT JEE. Students are recommended to download the Conic Section Notes for IIT JEE.
Conic Section is a curve generated by intersecting a right circular cone with a plane. Hyperbola, Parabola, Circle, and Ellipse are the basic four conic sections.
Table of contents
- Classification of Conic Section from its Equation
- Ellipse
- Parabola
- Hyperbola
- Circle
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Important terms in Conic Sections
(1) Directrix : The fixed straight line is called the directrix of the conic section.
(2) Focus : The fixed point is called the focus of the conic-section.
(3) Eccentricity : The constant ratio is called the eccentricity of the conic section and is denoted by e. If e>1, => hyperbola, e = 0 => circle, e = 1 => parabola and e <1 => ellipse.
(4) Axis: It is the straight line passing through the focus and perpendicular to the directrix. A conic is always symmetric about its axis.
(5) Latus Rectum: It is the chord passing through the focus and perpendicular to the major axis and also includes both endpoints on the curve.
(6) Vertex: It is the point of intersection of the conic section and the axis.
(7) Focal chord: A chord passing through the focus of the conic.
(8) Focal distance: It is the distance of any point on the conic from the focus.
(9) Double ordinate: The double ordinate of a conic is a chord perpendicular to the axis.
Classification Of Conic Section from its Equation
The equation of conics represented by the second degree equation can be written as ax^{2}+2hxy+by^{2}+2gx+2fy+c = 0. Here âˆ† = abc + 2fgh – af^{2} – bg^{2} – ch^{2}. âˆ† is the discriminant. We use this discriminant to determine the type of the conic section represented by the equation.
Condition |
Nature of Conic |
âˆ† â‰ 0, ab-h^{2} = 0 |
Parabola |
âˆ† â‰ 0, ab-h^{2} > 0 |
Ellipse |
âˆ† â‰ 0, ab-h^{2} < 0 |
Hyperbola |
âˆ† â‰ 0, a = b, h = 0 |
Circle |
âˆ† â‰ 0, ab-h^{2} < 0 and a+b = 0 |
Rectangular hyperbola |
How to find the Centre of a Conic
Let S be a conic given by ax^{2}+2hxy+by^{2}+2gx+2fy+c = 0.
Find dS/dx and dS/dy.
Solve dS/dx = 0 and dS/dy= 0 for x,y and get the centre (x,y).
(x, y) = ((hf-bg)/(ab-h^{2}), (gh-af)/(ab-h^{2}))
Ellipse
An ellipse is the locus of a point which moves in a plane so that the ratio of its distance from a fixed point and a fixed line is a constant which is less than 1. We call this ratio eccentricity and denote it by e.
Equation of Ellipse in Standard Form
The standard form equation of the ellipse is x^{2}/a^{2} + y^{2}/b^{2} = 1, (a>b).
Here a and b are constants.
Important Points to Remember:
- Length of major axis = 2a
- Length of minor axis = 2b
- Equation of Major axis is y = 0
- Equation of minor axis is x = 0
- Equation of directrix, x = +a/e
- Length of Latus Rectum = (2b^{2}/a)
- Eccentricity e = âˆš((a^{2}-b^{2})/a^{2})
- Sum of focal radii SP+Sâ€™P = 2a
- Distance between foci = 2ae
Parametric Form of the Ellipse
The equation of the ellipse in parametric form is given by x = a cos Î¸, y = b sin Î¸ where Î¸ is the eccentric angle.
Position of a point with respect to ellipse:
Let P(x_{1}, y_{1}) be any point and let x^{2}/a^{2 }+ y^{2}/b^{2} = 1 be the equation of an ellipse.
Let S_{1} = x_{1}^{2}/a^{2} + y_{1}^{2}/b^{2} – 1.
If S_{1}>0, the point lies outside the ellipse.
If S_{1} = 0, the point lies on the ellipse.
If S_{1}<0, the point lies inside the ellipse.
Equation of Tangent:
a) Point form: The equation of the tangent to the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1 at the point (x_{1}, y_{1}) is xx_{1}/a^{2} + yy_{1}/b^{2} = 1.
b) Parametric form: The equation of tangent at any point (a cos Î¸, b sin Î¸) is
(x/a) cos Î¸ + (y/b) sin Î¸ = 1
c) Slope form: Let the line y = mx+c touches the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1, then c^{2} = a^{2}m^{2}+b^{2}.
The straight line y = mxÂ±âˆš(a^{2}m^{2}+b^{2}) represents the tangents to the ellipse. The point of contact is ((Â±a^{2}m/âˆš(a^{2}m^{2}+b^{2}), Â± b^{2}/âˆš(a^{2}m^{2}+b^{2}))
Note:
a) The equation of the director circle of the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1 is x^{2} + y^{2} = a^{2} + b^{2}.
The director circle is the locus of the point of intersection of a pair of perpendicular tangents to an ellipse.
b) Two tangents can be drawn from a point to an ellipse. The two tangents are distinct and real or coincident or imaginary according as the given point lies outside, on or inside the ellipse.
c) The straight line lx + my + n = 0 touches the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1, if a^{2}l^{2} + b^{2}m^{2} = n^{2}.
d) The product of the perpendicular from the foci on any tangent to the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1 is equal to b^{2}.
Equation of pair of tangents:
Let P(x_{1}, y_{1}) be any point lying outside the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1 and let a pair of tangents PA, PB can be drawn to it from P. Then the equation of pair of tangents PA and PB is
SS_{1} = T^{2}
Where S = x^{2}/a^{2} + y^{2}/b^{2} – 1 = 0
S_{1} = x_{1}^{2}/a^{2} + y_{1}^{2}/b^{2} – 1 = 0
T = xx_{1}/a^{2} + yy_{1}/b^{2} – 1 = 0
Chord of Contact:
The equation of chord of contact of tangent drawn from a point P(x_{1}, y_{1}) to the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1 is T = 0 where
T = xx_{1}/a^{2} + yy_{1}/b^{2} – 1
Chord with a given Midpoint:
The equation of the chord of the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1 with P(x_{1}, y_{1}) as its middle point, is given by T = S_{1}.
T = xx_{1}/a^{2} + yy_{1}/b^{2} – 1
S_{1} = x_{1}^{2}/a^{2} + y_{1}^{2}/b^{2} – 1
Equation of Normal:
a) Point Form: The equation of the normal at (x_{1}, y_{1}) to the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1 is
a^{2}x/x_{1} – b^{2}y/y_{1} = a^{2 }– b^{2}.
b) Slope form: If m is the slope of the normal to the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1, then the equation of normal is y = mxÂ±m(a^{2} – b^{2})/âˆš(a^{2}+b^{2}m^{2})
The point of contact coordinates are (Â±a^{2}/âˆš(a^{2}+b^{2}m^{2}), Â±mb^{2}/âˆš(a^{2}+b^{2}m^{2}))
c) Parametric form: The equation of the normal to the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1 at (a cos Î¸, b sin Î¸) is ax sec Î¸ – by cosec Î¸ = a^{2} – b^{2}.
Tips to remember:
a) We can draw four normals from a point to an ellipse.
b) If y = mx + c is the normal of x^{2}/a^{2} + y^{2}/b^{2} = 1, then the condition for normality is
c^{2} = m^{2}(a^{2}-b^{2})^{2}/(a^{2}+b^{2}m^{2}).
c) The straight line lx + my + n = 0 is a normal to the ellipse x^{2}/a^{2} + y^{2}/b^{2} = 1, if
a^{2}/l^{2} + b^{2}/m^{2} = ((a^{2}-b^{2})/n^{2})^{2}.
d) Any point P of an ellipse is joined to the extremities of the major axis then the portion of a directrix intercepted by them subtends a right angle at the corresponding focus.
e) With a given point and line as focus and directrix, a series of ellipses can be described. The locus of the extremities of their minor axis is a parabola.
f) If S be the focus and G be the point where the normal at P meets the axis of an ellipse, then SG = e.SP and the tangent and normal at P bisect the external and internal angles between the focal distances of P.
Parabola
A parabola is the locus of a point whose distance from a focus (fixed point) is equal to the perpendicular distance from a fixed straight line (directrix).
Following are the four standard forms of the parabola.
- y^{2} = 4ax
- y^{2}= -4ax
- x^{2} = 4ay
- x^{2 }= -4ay
For parabola yÂ² = 4ax:
(a) Focus is (a, 0)
(b) Vertex is (0, 0)
(c) Axis is y = 0
(d) Directrix is x+a = 0
Focal Distance: The distance of a point on the parabola from the focus.
Focal Chord: A chord of the parabola, which passes through the focus.
Latus Rectum: A double ordinate passing through the focus or a focal chord perpendicular to the axis of the parabola is called the Latus Rectum (L.R.).
For y^{2} = 4ax, the length of the latus rectum = 4a.
Double Ordinate: A chord of the parabola perpendicular to the axis of the symmetry.
Note:
(i) Perpendicular distance from the focus on the directrix is equal to half the latus rectum.
(ii) Vertex is the middle point of the focus and the point of intersection of the directrix and axis.
(iii) Two parabolas are said to be equal if they have the same latus rectum.
Parabola and a Line
The line y = mx+c meets the parabola y^{2} = 4ax in
- two real points if a > mc.
- two coincident points if a = mc
- two imaginary points if a < mc
Condition of tangency is, c = a/m.
Length of the chord intercepted by the parabola on the line y = mx + c is (4/m^{2})âˆš(a(1+m^{2})(a-mc)).
Parametric Representation of Parabola
The parametric equation of parabola y^{2} = 4ax is given by x = at^{2}, y = 2at. So any point on this parabola is (at^{2}, 2at) which is called the t point.
Note:
(i) The equation of the chord joining t_{1} and t_{2} is 2x – (t_{1}+t_{2})y + 2at_{1}t_{2} = 0
(ii) Parametric equation of the parabola x^{2} = 4ay is x = 2at, y = at^{2}.
(iii) If t_{1} and t_{2} are the ends of a focal chord of the parabola y^{2} = 4ax, then t_{1}t_{2} = -1.
Chord of Contact
Equation of the chord of contact of tangents drawn from a point P(x_{1}, y_{1}) is
yy_{1} = 2a(x+x_{1}) (ie. T = 0)
Chord with a Given Midpoint
The equation of the chord of the parabola y^{2} = 4ax whose middle point is (x_{1}, y_{1}) is
(y-y_{1}) = (2a/y_{1})(x-x_{1}) â‰¡ T â‰¡ S_{1}.
Director circle
It is the locus of the point of intersection of the perpendicular tangents to a curve. For parabola y^{2} = 4ax, the equation of the director circle is x + a = 0 which is parabolaâ€™s own directrix.
Position of a point relative to a parabola
The point (x_{1}, y_{1}) lies outside the parabola y^{2} = 4ax, if y_{1}^{2}-4ax_{1} is positive.
The point (x_{1}, y_{1}) lies on the parabola y^{2} = 4ax, if y_{1}^{2}-4ax_{1} is zero.
The point (x_{1}, y_{1}) lies inside the parabola y^{2} = 4ax, if y_{1}^{2}-4ax_{1} is negative.
Equation of tangent to the parabola y^{2} = 4ax :
a) Point form: yy_{1} = 2a(x+x_{1}) at the point (x_{1}, y_{1}).
b) Parametric form: ty = x+at^{2} at point (at^{2}, 2at)
c) Slope form: y = mx+a/m (m â‰ 0) at point (a/m^{2}, 2a/m)
Equation of normal to the parabola y^{2} = 4ax:
a) Point form: y – y_{1} = (-y_{1}/2a)(x-x_{1}) at the point (x_{1}, y_{1}).
b) Parametric form: y+tx = 2at+at^{3} at point (at^{2}, 2at)
c) Slope form: y = mx-2am-am^{3 }at point (am^{2}, -2am)
Note:
a) Condition for normality: The line y =mx+c is a normal to the parabola y^{2} = 4ax if
c = -2am – am^{3} and x^{2} = 4ay if c = 2a + a/m^{2}.
b) The foot of the normal is (am^{2}, -2am)
Pair Of Tangents
The equation to the pair of tangents which can be drawn from any point (x_{1}, y_{1}) to the parabola y^{2} = 4ax is given by:
SS_{1} â‰¡ T^{2} where : S â‰¡ y^{2}-4ax
S_{1} = y_{1}^{2} – 4ax_{1}
T â‰¡ yy_{1} – 2a(x+x_{1})
Hyperbola
The conic whose eccentricity is greater than unity (e > 1) is a hyperbola.
Standard Equation:
The standard equation of a hyperbola is x^{2}/a^{2} – y^{2}/b^{2} = 1 where b^{2} = a^{2}(e^{2} – 1).
Eccentricity (e): e^{2} = 1 + b^{2}/a^{2} = 1 + (CA/TA)^{2}
Conjugate axis â†’ CA
Transverse axis â†’ TA
Foci: S = (ae, 0) and Sâ€™ = (-ae, 0)
Equations of Directrix : x = a/e, and x = -a/e.
Principal Axes: The transverse axis and the conjugate axis together are called principal axes of the hyperbola.
Vertices : A â‰¡ (a, 0) and A’ â‰¡ (-a, 0)
Focal Chord : A chord which passes through a focus is called a focal chord.
Double Ordinate : A chord perpendicular to the transverse axis is called a double ordinate.
Latus Rectum : The focal chord perpendicular to the transverse axis is called the latus rectum.
l = 2b^{2}/a = 2a(e^{2} – 1)
Rectangular hyperbola: The hyperbola in which the lengths of the transverse and conjugate axis are equal is called a rectangular hyperbola. The eccentricity of the rectangular hyperbola is âˆš2.
Conjugate hyperbolas: Two hyperbolas are conjugate hyperbolas of each other if the transverse axis and conjugate axes of one hyperbola are respectively the conjugate and the
transverse axes of the other.
x^{2}/a^{2} – y^{2}/b^{2} = 1 and -x^{2}/a^{2} + y^{2}/b^{2} = 1 are conjugate hyperbolas of each other.
Hyperbola and a Line
The line y = mx+c is a secant to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 if c^{2} > a^{2}m^{2} – b^{2}.
The line is a tangent to the hyperbola if c^{2} = a^{2}m^{2} – b^{2}.
The line passes outside the hyperbola, if c^{2} < a^{2}m^{2} – b^{2}.
Parametric Representation of Hyperbola
The equation x = a sec Î¸, y = b tan Î¸ represents the parametric equation of the hyperbola
x^{2}/a^{2} – y^{2}/b^{2} = 1, where Î¸ is the parameter (Î¸ is called the eccentric angle of the point P).
Chord of Contact
The equation of the chord of contact of tangent drawn from a point P(x_{1}, y_{1}) to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 is T = 0, where T â‰¡ xx_{1}/a^{2} – yy_{1}/b^{2} – 1.
Chord with a Given Midpoint
The equation of chord of the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 with P(x_{1}, y_{1}) as its middle point is given by T = S_{1}, where T â‰¡ xx_{1}/a^{2} – yy_{1}/b^{2}_{ }– 1 and S_{1}â‰¡ x_{1}^{2}/a^{2} – y_{1}^{2}/b^{2} – 1.
Position of a point relative to a hyperbola
The point P(x_{1}, y_{1}) lies outside the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1, if x_{1}^{2}/a^{2} – y_{1}^{2}/b^{2} – 1 < 0.
The point P(x_{1}, y_{1}) lies inside the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1, if x_{1}^{2}/a^{2} – y_{1}^{2}/b^{2} – 1 > 0.
The point P(x_{1}, y_{1}) lies on the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1, if x_{1}^{2}/a^{2} – y_{1}^{2}/b^{2} – 1 = 0.
Equation of Tangent
1) Point form:
The equation of tangent to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 at the point (x_{1}, y_{1}) is xx_{1}/a^{2} – yy_{1}/b^{2} = 1.
2) Slope form: The equation of tangent to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 in terms of slope m is
y = mxÂ±âˆš(a^{2}m^{2}-b^{2})
The coordinates of the points of contact are (Â±a^{2}m/âˆš(a^{2}m^{2}-b^{2}), Â±b^{2}/âˆš(a^{2}m^{2}-b^{2})).
3) Parametric form: Equation of the tangent to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 at the point
(a sec Î¸, b tan Î¸) is (x/a) sec Î¸ – (y/b) tan Î¸ = 1
Note:
a) There are two parallel tangents having the same slope m. These tangents touch the hyperbola at the extremities of a diameter
b) If |Î¸_{1} + Î¸_{2}| = Ï€, then the tangents at these points Î¸_{1} and Î¸_{2 }are parallel.
Condition for tangency and points of contact
The condition for the line y = mx+c to be a tangent to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 is that
c^{2} = a^{2}m^{2} – b^{2} and the coordinates of the points of contact are (Â±a^{2}m/âˆš(a^{2}m^{2}-b^{2}), Â±b^{2}/âˆš(a^{2}m^{2}-b^{2}).
Equation of the Pair of Tangents
The equation of the pair of tangents drawn from a point P(x_{1}, y_{1}) to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 is given by SS_{1} = T^{2},
where S â‰¡ x^{2}/a^{2} – y^{2}/b^{2} – 1, S_{1} â‰¡ x_{1}^{2}/a^{2} – y_{1}^{2}/b^{2} – 1 and T â‰¡ xx_{1}/a^{2} – yy_{1}/b^{2} – 1.
Equation of Normal
1) Point form:
The equation of normal to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 at the point P(x_{1}, y_{1}) on it is
(a^{2}x/x_{1}) + (b^{2}y/y_{1}) = a^{2}+b^{2} = a^{2}e^{2}
2) Parametric form:
The equation of normal to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 at the point P(a sec Î¸, b tan Î¸) is
ax/sec Î¸ + by/tan Î¸ = a^{2} + b^{2} = a^{2}e^{2}
3) Slope form:
The equation of normal to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 in terms of slope m is
y = mx – (a^{2}+b^{2})m/âˆš(a^{2}-b^{2}m^{2}).
4) Condition for normality
If y = mx+c is the normal of x^{2}/a^{2} – y^{2}/b^{2} = 1, then condition for normality is
c = Â±m(a^{2}+b^{2})/âˆš(a^{2}-m^{2}b^{2}) or
c^{2} = m^{2}(a^{2}+b^{2})^{2}/(a^{2}-m^{2}b^{2})
5) Points of contact: Coordinates of point of contact are (Â±a^{2}/âˆš(a^{2}-m^{2}b^{2}), Â±mb^{2}/âˆš(a^{2}-m^{2}b^{2})).
Note:
a) If the line lx+my+n = 0 will be normal to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1,
then a^{2}/l^{2} – b^{2}/m^{2} = (a^{2}+b^{2})^{2}/n^{2}
b) In general four normals can be drawn to a hyperbola from any point and if Î±, Î², Î³, Î´ be the eccentric angles of these four co-normal points, then Î± + Î² + Î³ + Î´ is an odd multiple of Ï€.
c) If the normal at P meets the transverse axis at G, then SG = e.SP. Also the tangent and normal bisect the angle between the focal distances of P.
Director Circle:
It is the locus of the intersection point of tangents which are at right angles.
The equation to the director circle is given by x^{2}+y^{2} = a^{2}-b^{2} .
The circle is real if b^{2} < a^{2} .
If b^{2} = a^{2}, the radius of the circle is zero and it reduces to a point circle at the origin.
If b^{2} > a^{2} , the radius of the circle is imaginary, so that there is no such circle and so no pair of tangents at right angles can be drawn to the curve.
Circle
A locus of a point which moves at a constant distance from a fixed point is called a circle. The constant distance is called the radius of the circle and the fixed point is called the centre of the circle.
General Equation of a Circle:
The general equation of a Circle is x^{2}+y^{2}+2gx+2fy+c = 0, g, f and c are constants.
Centre of the Circle = (-g, -f)
Radius of a Circle = âˆš(g^{2}+f^{2}-c)
The equation of a circle with centre (h, k) and radius r is (x-h)^{2}+(y-k)^{2} = r^{2}.
If origin is the centre of the circle, then the equation becomes x^{2}+y^{2} = r^{2}.
If r = 0, then the circle is called the point circle and its equation is (x-h)^{2}+(y-k)^{2} = 0.
Parametric Equation of Circle
a) The parametric equation of a circle x^{2}+y^{2} = r^{2} is x = r cos Î¸, y = r sin Î¸.
So the parametric coordinates of any point lying on the circle x^{2} + y^{2} = r^{2} are (r cos Î¸, r sin Î¸)..
b) The parametric equation of a circle (x-h)^{2}+(y-k)^{2} = r^{2} is x = h + r cos Î¸, y = k + r sin Î¸.
So the parametric coordinates of any point lying on the circle are (h + r cos Î¸, k + r sin Î¸).
Position of a Point with respect to a Circle
A point (x_{1},y_{1}) lies outside the circle S â‰¡ x^{2}+y^{2}+2gx+2fy+c = 0 if S_{1} â‰¡ x_{1}^{2}+y_{1}^{2}+2gx_{1}+2fy_{1}+c is positive (S_{1} > 0).
If S_{1} < 0, the point lies inside the circle.
If S_{1} = 0, the point lies on the circle.
Line and a Circle
Let L = 0 be a line and S = 0 be a circle, if r be the radius of a circle and p be the length of the perpendicular from the centre of circle to the line, then if
p=r, => Line touches the circle.
p>r, => Line is outside the circle.
p<r, => Line is the chord of the circle.
p=0, => Line is the diameter of the circle.
Condition for Tangency
A line L = 0 touches the circle S = 0, if the length of perpendicular drawn from the centre of the circle to the line is equal to the radius of the circle. i.e p = r. This is the condition of tangency for the line L = 0.
The circle x^{2}+y^{2} = a^{2} touches the line y = mx+c if c = Â±aâˆš(1+m^{2})
a) If a^{2}(1+m^{2}) – c^{2} >0, the line will meet the circle at real and different points.
b) If c^{2} = a^{2}(1+m^{2}), the line will touch the circle.
c) If a^{2}(1+m^{2}) – c^{2} <0, the line will meet the circle at two imaginary points.
Equation of Tangent
The equation of tangent to the circle x^{2}+y^{2}+2gx+2fy+c = 0 at a point (x_{1}, y_{1}) is
xx_{1}+yy_{1}+g(x+x_{1})+f(y+y_{1})+c = 0 or T = 0.
Note:
a) The equation of tangent to the circle x^{2}+y^{2} = a^{2} at the point (x_{1}, y_{1}) is
xx_{1}+yy_{1}=a^{2}.
b) From condition of tangency for every value of m, the line y = mxÂ±aâˆš(1+m^{2}) is a tangent of the circle x^{2}+y^{2} =a^{2} and its point of contact is (Â±am/âˆš(1+m^{2}), Â±a/âˆš(1+m^{2}))
Equation of Normal
Normal to a curve at any point P of a curve is the straight line that passes through P and is perpendicular to the tangent at P.
The equation of normal to the circle x^{2}+y^{2}+2gx+2fy+c = 0 at a point (x_{1}, y_{1}) is
y-y_{1} = (y_{1}+f)(x-x_{1})/(x_{1}+g)
Note:
a) The equation of normal to the circle x^{2}+y^{2} = a^{2} at the point (x_{1}, y_{1}) is
xy_{1}-x_{1}y = 0.
b) The equation of any normal to the circle x^{2}+y^{2} = a^{2} is y = mx where m is the slope of the normal.
c) The normal to the circle will pass through the centre of the circle.
d) The equation of any normal to the circle x^{2}+y^{2}+2gx+2fy+c = 0 is y+f = m(x+g), where m is the slope of the normal.
Length of Tangent
Two tangents can be drawn from a point P(x_{1}, y_{1}) to a circle. Let PQ and PR be two tangents drawn from P(x_{1}, y_{1}) to the circle x^{2}+y^{2}+2gx+2fy+c = 0. Then PQ = PR is called the length of the tangent drawn from point and is given by
PQ=PR=âˆš(x_{1}^{2}+y_{1}^{2}+2gx_{1}+2fy_{1}+c) = âˆšS_{1}
Pair of Tangents
Two tangents PQ and PR can be drawn from a point P(x_{1}, y_{1}) to the circle S = x^{2}+y^{2}+2gx+2fy+c = 0.
The combined equation is SS_{1} = T^{2},
where S = 0 is the equation of circle, T = 0 is the equation of tangent at (x_{1}, y_{1}) and S_{1} is obtained by replacing x by x_{1} and y by y_{1} in S.
Director Circle:
It is the locus of the point of intersection of two perpendicular tangents to a circle.
Let the circle be x^{2}+y^{2}=a^{2}, then equation of the pair of tangents to a circle from a point (x_{1}, y_{1}) is (x^{2}+y^{2}-a^{2})(x_{1}^{2}+y_{1}^{2}-a^{2}) = (xx_{1}+yy_{1}-a^{2})^{2}.
If this represents a pair of perpendicular lines then coefficient of x^{2} + coefficient of y^{2} = 0
The equation of the director circle is x^{2}+y^{2} = 2a^{2}.
Director circle is a concentric circle whose radius is âˆš2 times the radius of a given circle.
Chord of Contact
The chord joining the two points of contact of tangents to a circle drawn from any point P is called the chord of contact of P with respect to the given circle.
Let the given point is P(x_{1}, y_{1}) and the circle is S = 0 then the equation of the chord of contact is T = xx_{1}+yy_{1}+g(x+x_{1})+f(y+y_{1})+c = 0
The length of the chord of contact is 2âˆš(r^{2}-p^{2}).
If the point (x_{1}, y_{1}) lies on the circle, then the chord of contact coincides with the equation of the tangent.
Equation of Chord whose midpoint is given
The equation of the chord of the circle x^{2}+y^{2} = a^{2} whose midpoint P(x_{1}, y_{1}) is given is
y-y_{1} = (-x_{1}/y_{1})(x-x_{1}) or xx_{1}+yy_{1} = x_{1}^{2}+y_{1}^{2}, which can be represented by T = S_{1}.
Common Chord of two Circles
The line joining the points of intersection of two circles is called the common chord.
Let S_{1} and S_{2} be the 2 circles
S_{1} = x^{2}+y^{2}+2g_{1}x+2f_{1}x+c_{1} = 0 and S_{2}= x^{2}+y^{2}+2g_{2}x+2f_{2}x+c_{2} = 0
Then the equation of common chord is S_{1}-S_{2} = 0
=> 2x(g_{1}-g_{2}) + 2y(f_{1}-f_{2}) + c_{1}-c_{2} = 0
Note:
a) Two circles touch each other if the length of their common chord is zero.
b) Maximum length of the common chord = diameter of the smaller circle.
Important Points to Remember
a) If r is the radius of the circle, the diameter d = 2r is the maximum distance between two points on the given circle.
b) The area of the circle = Ï€r^{2} or Ï€d^{2}/4.
c) Angle subtended at the centre of a circle by any arc = arc/radius.
d) The circumference of the circle = 2Ï€r or Ï€d.
e) Angle subtended at the centre of a circle by an arc is double of angle subtended at the circumference of a circle.
Solved Examples
Question 1:
The equation of the conic with focus at (1, -1), directrix along x-y+1 = 0 and with eccentricity âˆš2 is
a) x^{2}-y^{2} = 1
b) xy = 1
c) 2xy-4x+4y+1 = 0
d) 2xy+4x-4y-1 = 0
Solution:
Let P(x, y) be any point on the conic.
SP = e. PM
Focus, S = (1, -1)
âˆš[(x-1)^{2}+(y-1)^{2}] = âˆš2(x-y+1)/âˆš2
=> (x-1)^{2}+(y-1)^{2 }= (x-y+1)^{2}
2xy-4x+4y+1 = 0
Hence option c is the answer.
Question 2:
The foci of the hyperbola 9x^{2}-16y^{2} = 144 is
a) (Â±4, 0)
b) (0, Â±4)
c) (0, Â±5)
d) (Â±5, 0)
Solution:
Given 9x^{2}-16y^{2} = 144
=> x^{2}/16 – y^{2}/9 = 1
a = 4, b = 3
b^{2} = a^{2}(e^{2}-1)
=> 9 = 16(e^{2}-1)
=> e = 5/4.
So foci are (Â±ae, 0)
=> (Â±5, 0)
Hence option d is the answer.
Question 3:
If y = 2x+k is a diameter to the circle 2(x^{2}+y^{2})+3x+4y-1 = 0, then value of k is
a) 0
b) 1
c) 2
d) 1/2
Solution:
Given circle 2(x^{2}+y^{2})+3x+4y-1 = 0
Comparing with x^{2}+y^{2}+2gx+2fy+c = 0, we get g = 3/4, f = 1
Centre = (-g, -f)
=> centre = (-3/4, -1)
Centre lies on the diameter of the circle.
y = 2x+k
Substitute (-3/4, -1) in above equation
=> -1 = -3/2 + k
=> k = -1+3/2
= 1/2
Hence option d is the answer.
Question 4:
Equation of line passing through the points of intersection of the parabola x^{2} = 8y and the ellipse x^{2}/3 + y^{2} = 1 is
a) y+3=0
b) 3y+1 = 0
c) 3y-1 = 0
d) y-3 = 0
Solution:
Given parabola x^{2} = 8y â€¦(i)
Ellipse x^{2}/3 + y^{2} = 1â€¦(ii)
Put (i) in (ii)
=> 8y/3 + y^{2} = 1
=> 3y^{2}+8y-3 = 0
=> (3y-1)(y+3) = 0
=> y = 1/3 and y = -3
When y = -3, x^{2} = -24 (not possible)
So y = 1/3.
Put y in (i)
=> x^{2} = 8/3
=> x = Â±âˆš(8/3)
Point of intersection is (âˆš(8/3), 1/3) and (-âˆš(8/3), 1/3).
Equation of required line is y = 1/3
=> 3y – 1 = 0
Hence option c is the answer.
Question 5:
Let L be a normal to the parabola y^{2} = 4x. If L passes through the point (9, 6), then L is given by
a) y-x+3=0
b) y+3x-33=0
c) y+x-15=0
d) y-2x+12=0
Solution:
Given parabola is y^{2} = 4x
Differentiate w.r.t.x
=> 2y (dy/dx) = 4
(dy/dx) = 4/2y
= 2/y
Slope of tangent at (9, 6) => 2/6 = 1/3
Slope of normal = -3
Equation of the normal at (9, 6) is y – 6 = -3(x – 9)
y + 3x – 33 = 0
Hence option b is the answer.
Practice Problems
1. The ends of the latus rectum of a parabola are (-3,1) and (1,1). Then the equation of the parabola is:
a) (x+1)^{2} = 4y
b) (x-1)^{2} = 4y
c) (x+1)^{2} = 2y
d) (x-1)^{2} = 2y
2. The equation of a conic section whose focus is at (-1,0), directrix is the line 4x-3y+2=0 and eccentricity is 1/âˆš2 is
a) 34x^{2}+24xy+41y^{2}+84x+12y+46 = 0
b) 41x^{2}+24xy+34y^{2}+84x+12y+46 = 0
c) 34x^{2}+24xy+41y^{2}+84x+12y+81 = 0
d) 34x^{2}+24xy+41y^{2}+12x+84y+46 = 0
3. Two common tangents to the circle x^{2}+y^{2} = 2a^{2} and parabola y^{2} = 8ax are
a) x = Â±(y+2a)
b) y = Â±(x+2a)
c) x = Â±(y+a)
d) y = Â±(x+a)
4. The eccentricity of an ellipse, with its centre at the origin is 1/2. If one of the directrices is x = 4, then the equation of the ellipse is
a) 3x^{2}+4y^{2} = 1
b) 3x^{2}+4y^{2} = 12
c) 4x^{2}+3y^{2} = 12
d) 4x^{2}+3y^{2} = 1
5. The locus of a point P(Î±, Î²) moving under the condition that the line y = Î±x+ Î² is a tangent to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 is
a) an ellipse
b) a circle
c) a parabola
d) a hyperbola
Frequently Asked Questions
What are the 4 conic sections?
Depending upon the inclination of angle between the plane and the cone, conic sections can be divided into 4 types. Circles, Ellipses, Parabolas, Hyperbolas.
What is a rectangular hyperbola?
Rectangular hyperbola is a hyperbola in which the lengths of the transverse and conjugate axis are equal. The eccentricity of the rectangular hyperbola is âˆš2.
How is an ellipse formed?
An ellipse is formed by a plane intersecting a cone at an angle to its base.
What is the definition of a conic section?
A conic section is defined as the curve formed from a planeâ€™s intersection with a right circular cone.
Who discovered conic sections?
Menaechmus, an ancient Greek mathematician discovered the conic sections.
What is the importance of conic sections in real life?
Conics are used to explain the orbits of planets, moons and satellites in our universe. Buildings, bridges, and statues use conics as support systems.