# Tangent To a Parabola

## Trending Questions

**Q.**

A line parallel to the straight line $2x-y=0$ is tangent to the hyperbola $\frac{{x}^{2}}{4}-\frac{{y}^{2}}{2}=1$ at the point$({x}_{1},{y}_{1})$.

Then ${{x}_{1}}^{2}+5{{y}_{1}}^{2}$ is equal to

$6$

$10$

$8$

$5$

**Q.**If the tangent to the parabola y2=x at a point (α, β), (β>0) is also a tangent to the ellipse, x2+2y2=1, then α is equal to :

- 2√2−1
- 2√2+1
- √2+1
- √2−1

**Q.**The equation of common tangent to the curves y2=16x and xy= –4, is :

- x−2y+16=0
- x+y+4=0
- 2x−y+2=0
- x−y+4=0

**Q.**A line is a common tangent to the circle (x–3)2+y2=9 and the parabola y2=4x. If the two points of contact (a, b) and (c, d) are distinct and lie in the first quadrant, then 2(a+c) is equal to

**Q.**Equation of a common tangent to the parabola y2=4x and the hyperbola xy=2 is:

- x−2y+4=0
- x+y+1=0
- 4x+2y+1=0
- x+2y+4=0

**Q.**The slope of the line touching both the parabolas y2=4x and x2=−32y is

- 12
- 32
- 18
- 23

**Q.**A parabola y=ax2+bx+c crosses the x-axis at (α, 0)(β, 0) both to the right of the origin. A circle also passes through these two points. The length of a tangent from the origin to the circle is

- √bca
- √ca
- ac2
- ba

**Q.**The equation of the latus rectum of a parabola is x+y=8 and the equation of the tangent at the vertex is x+y=12. Then the length of the latus rectum is

- 8 units
- 8√2 units
- 2√2 units
- 4√2 units

**Q.**If the tangents to the parabola x=y2+c from origin are perpendicular, then c is equal to

- 12
- 1
- 14
- 2

**Q.**Tangents are drawn from the point (−8, 0) to the parabola y2=8x touch the parabola at P and Q. If F is the focus of the parabola, then the area of the triangle PFQ (in sq. units) is equal to

- 64
- 24
- 32
- 48

**Q.**Consider the parabola y2=8x. Let Δ1 be the area of the triangle formed by the endpoints of its latus rectum and the points P(12, 2) on the parabola and Δ2 be the area of the triangle formed by drawing tangents at P and at the endpoints of the latus rectum. Then, Δ1Δ2 is

**Q.**Two parabolas with a common vertex and with axes along x− axis and y− axis, respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is :

- 4(x+y)+3=0
- 3(x+y)+4=0
- x+2y+3=0
- 8(2x+y)+3=0

**Q.**

The portion of a tangent to a parabola cut off between the directrix and the curve subtends right angle at the focus.

True

False

**Q.**The straight line x+y=k+1 touches the parabola y=x(1−x) if

- k=−1
- k=0
- k=1
- k takes any real value

**Q.**

The equation of four circles are (x±a)2+(y±a)2=a2 . The radius of a circle touching all the four circles is

None of these

**Q.**The equation(s) of tangents to the parabola y2 = 12x, which passes through the point (2, 5) is/are:

- y = x + 3
- y = x - 3
- 2y = 3x - 4
- 2y = 3x + 4

**Q.**Consider two concentric circles C1:x2+y2−4=0 and C2:x2+y2−9=0. A parabola is drawn through the points where C1 meets y−axis and having an arbitrary tangent of C2 as its directrix. If C is the curve of locus of focus of drawn parabola and e is the eccentricity of the curve C, then the value of 12e is

**Q.**The tangent to the parabola y=x2+ax+1 at the point of intersection of y−axis also touches the circle x2+y2=r2 and no point of the parabola is below x−axis. Then

- the radius of circle when ′a′ attains its maximum value is 1√10
- the slope of the tangent when radius of the circle is maximum is 0
- area bounded by the tangent and the coordinate axes is minimum when a=1
- the minimum area bounded by the tangent and the coordinate axes is 14 sq. unit

**Q.**Consider the parabola (x−1)2+(y−2)2=(12x−5y+3)2169

Column IColumn IIa. Locus of point of intersection of perpendicular tangent p. (12x−5y−2=0) b. Locus of foot of perpendicular from focus upon any tangent q. (5x+12y−29=0)c. Line along which minimum length of focal chord occurs r. (12x−5y+3=0) d. Line about which parabola is symmetrical s. (24x−10y+1=0)

- a−r, b−s, c−p, d−q
- a−s, b−r, c−p, d−q
- a−r, b−q, c−p, d−s
- a−p, b−s, c−q, d−r

**Q.**The line x+y=1 touches the parabola y2−y+x=0 at the point

- (1, 1)
- (12, 12)
- (0, 1)
- (1, 0)

**Q.**Two straight lines are perpendicular to each other. One of them touches the parabola y2=4a(x+a) and the other touches y2=4b(x+b). Their point of intersection lies on the line

- x−a+b=0
- x+a−b=0
- x+a+b=0
- x−a−b=0

**Q.**

Let $P$ and $Q$ are the points on the parabola ${y}^{2}=4x,$ if so that the line segment $PQ$ subtends a right angle at the vertex. If $PQ$ intersects the axis of the parabola at $R$, then the distance of the vertex from $R$ is

$1$

$2$

$4$

$6$

**Q.**The area of the region bounded by the curve y=x3, its tangent at (1, 1) and x-axis is

- 112
- 217
- 16
- 215

**Q.**If tangents are drawn at the points A(1, 2), B(4, −4) and a variable point ′C′ lies on the parabola y2=4x, then the locus of the orthocentre of triangle formed by these tangents is

- x=0
- x−1=0
- x+1=0
- y=0

**Q.**

The curve given by $x+y={e}^{xy}$has a tangent parallel to the$y$- axis at the point :

$\left(0,1\right)$

$\left(1,1\right)$

$\left(0,0\right)$

$\left(1,0\right)$

**Q.**The common tangents to the circle x2+y2=2 and the parabola y2=8x touch the circle at the points P, Q and the parabola at the points R, S. Then the area of the quadrilateral PQRS is

- 3
- 6
- 9
- 15

**Q.**If α is the inclination of a tangent to the parabola y2=4ax then the distance between the tangent and a parallel normal is

**Q.**The equation(s) of tangents to the parabola y2 = 12x, which passes through the point (2, 5) is/are:

- y = x + 3
- y = x - 3
- 2y = 3x - 4
- 2y = 3x + 4

**Q.**If a chord of solpe m of the circle x2+y2=4 touches the parabola y2=4x, then

(where [.] represents G.I.F)

- least value of [m2]=0
- least value of [m]=0
- m∈⎛⎝−∞, −√√2−12⎞⎠∪⎛⎝√√2−12, ∞⎞⎠
- m∈⎛⎝−√√2−12, √√2−12⎞⎠

**Q.**The equation of common tangent to the curves y2=16x and xy= –4, is :

- x−2y+16=0
- x+y+4=0
- 2x−y+2=0
- x−y+4=0