A curve in the plane which surrounds the 2 focal points such that the total of the distances to the focal point remains constant for each point on the curve. A circle is said to be a special type of an ellipse having both focal points at the same point. A line which intersects the ellipse at a point is called a tangent to the ellipse. The different forms of the tangent equation are given below:
- Slope form of a tangent to an ellipse
If the line y = mx + c touches the ellipse x2 / a2 + y2 / b2 = 1, then c2 = a2m2 + b2. The straight line y = mx ∓ √[a2m2 + b2] represent the tangents to the ellipse.
- Point form of a tangent to an ellipse
The equation of the tangent to an ellipse x2 / a2 + y2 / b2 = 1 at the point (x1, y1) is xx1 / a2 + yy1 / b2 = 1.
- Parametric form of a tangent to an ellipse
The equation of the tangent at any point (a cosɸ, b sinɸ) is [x / a] cosɸ + [y / b] sinɸ.
- Point of contact of the tangent to an ellipse
Line y = mx ∓ √[a2m2 + b2] touches the ellipse x2 / a2 + y2 / b2 = 1 at (∓a2m / √[a2m2 + b2]) , (∓b2 / √[a2m2 + b2]).
Equation Of Tangent To Ellipse Problems
Example 1: What is the locus of the point of intersection of perpendicular tangents to the ellipse x2 / a2 + y2 / b2 = 1?
Let point be (h,k). The pair of tangents will be
(x2 / a2 + y2 / b2 − 1) (h2 / a2 + k2 / b2 − 1) = (hx / a2 +yk / b2 − 1)2
Pair of tangents will be perpendicular, if coefficient of x2 + coefficient of y2 = 0
k2 / a2b2 + h2 / a2b2 = 1 / a2 + 1 / b2
h2 + k2 = a2 + b2
Replace (h, k) by (x, y)
x2 + y2 = a2 + b2.
Example 2: What is the condition for the line lx + my − n = 0 to be tangent to the ellipse x2 / a2 + y2 / b2 = 1?
y = [−l / m] x + n / m is tangent to x2 / a2 + y2 / b2 = 1, if
n / m = ± √b2 + [a2 / (l / m)2] or
n2 = m2b2 + l2a2.
Example 3: If any tangent to the ellipse x2 / a2 + y2 / b2 = 1 cuts off intercepts of length h and k on the axes, then a2 / h2 + b2 / k2 = ___________.
The tangent at (a cosθ, b sinθ) to the ellipse is [(a cos θ) * x] / [a2] +[b sin θ] * y / b2 = 1
or x / (a / cos θ) + y / (b / sin θ) = 1
∴Intercepts are, h = a / cos θ, k = b sin θ
a2 / h2 + b2 / k2 = 1
Example 4: The equation of the tangents drawn at the ends of the major axis of the ellipse 9x2 + 5y2 − 30y = 0, are ___________.
Change the equation 9x2 + 5y2 − 30y = 0 in standard form 9x2 + 5 (y2 − 6y) = 0
9x2 + 5 (y2 − 6y + 9) = 45
x2 / 5 + (y − 3)2 / 9 = 1
∵ a2 < b2, so axis of the ellipse is on the y-axis.
At y-axis, put x = 0, so we can obtain the vertex.
Then 0 + 5y2 − 30y = 0
y = 0, y = 6
Therefore, tangents of vertex y = 0, y = 6.
Example 5: The equation of tangent and normal at point (3, 2) of ellipse 4x2 + 9y2 = 36 are _________.
Given, equation of an ellipse is 4x2 + 9y2 = 36
Tangent at point (3, 2) is (3) * x / 9 + (−2) * y / 4 = 1 or x / 3 − y / 2 = 1
∴Normal is x / 2 + y / 3 = k and it passes through the point (3,2)
∴ 3 / 2 − 2 / 3 = k ⇒ k = 5 / 6
∴ Normal is, x / 2 + y / 3 = 5 / 6.
Example 6: Minimum area of the triangle by any tangent to the ellipse x2 / a2 + y2 / b2 = 1 with the coordinate axes is _____________.
Equation of tangent at (a cos θ, b sin θ) is [x / a] cos θ + [y / b] sin θ = 1
P = (a / cos θ, 0)
Q = (0, b / sin θ)
Area of OPQ = 1 / 2 ∣(a / cos θ) (b / sin θ)∣ = ab / |sin 2θ|
(Area)min = ab
Example 7: The eccentric angles of the extremities of latus recta of the ellipse
Coordinates of any point on the ellipse
whose eccentric angle is
The coordinates of the end points of latus recta are
Example 8: The length of the axes of the conic
Given that, the equation of conic
Length of axes are
Example 9: The coordinates of the foci of the ellipse
A) (1, 2), (3, 4)
B) (1, 4), (3, 1)
C) (1, 1), (3, 1)
D) (2, 3), (5, 4)