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Parametric Equation of Tangent

Let tangent d...

Question

Let tangent drawn to the parabola $C_{1} : y^{2} = 4 a x$ at $P$ meets the $y$ -axis at $Q$ . Another parabola $C_{2}$ with vertex $Q$ and focus $(0, 0)$ is drawn which passes through $(2 a, 0) .$ Identify which of the following statements can be CORRECT?

The equation of the curve

C_{2}

x^{2} = 4 a (y + a)

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The equation of the curve

C_{2}

x^{2} = - 4 a (y + a)

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Point

P

is the extremity of latus rectum of curve

C_{1}

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Possible equation of directrix of curve

C_{2}

y - 2 a = 0

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Solution

The correct option is D Possible equation of directrix of curve $C_{2}$ is $y - 2 a = 0$
Tangent at $P (a t^{2}, 2 a t)$ is
$t y = x + a t^{2}$
$Q \equiv (0, a t)$

Now, equation of parabola with vertex $(0, a t)$ and $(0, 0)$ as its focus is $x^{2} = - 4 a t (y - a t)$
The curve passes through $(2 a, 0)$ $4 a^{2} = - 4 a t (- a t)$
$\Rightarrow t^{2} = 1 \Rightarrow t = \pm 1$

Possible equation of curve $C_{2}$ is
$x^{2} = - 4 a (y - a)$
or $x^{2} = 4 a (y + a)$
Co-ordinates of $P$ are $(a, \pm 2 a)$
$[$ Extremities of latus rectum of $C_{1}]$

Possible equation of directrix of $C_{2} :$
$y - a = a \Rightarrow y - 2 a = 0$
or $y + a + a = 0 \Rightarrow y + 2 a = 0$

Suggest Corrections

Similar questions

Q. Let tangent drawn to the parabola

C_{1} : y^{2} = 4 a x

P

meets the

y

-axis at

Q

. Another parabola

C_{2}

with vertex

Q

and focus

(0, 0)

is drawn which passes through

(2 a, 0) .

Identify which of the following statements can be CORRECT?

Q. From a point

P

on the circle

C_{1} : x^{2} + y^{2} = 25

, tangents are drawn to the circle

C_{2} : x^{2} + y^{2} = 9

. Let

θ

be the angle made by the line

P O

(

O

is the origin) with positive

x

-axis. If the chord of contact of the circle

C_{2}

is tangent to another circle

C_{3}

which passes through the origin, then locus of center of the circle

C_{3}

Q. Let normal's drawn to parabola at point's

P (0, 0)

and

Q (3, - 1)

intersect at

(2, 1)

. If

P Q

is bisected by the axis of the parabola, then

Q. A parabola is drawn with focus at

(3, 4)

and vertex at the focus of another parabola

y^{2} - 12 x - 4 y + 4 = 0

. Then equation of the parabola is

Q. Consider two concentric circles

C_{1} : x^{2} + y^{2} - 4 = 0

and

C_{2} : x^{2} + y^{2} - 9 = 0.

A parabola is drawn through the points where

C_{1}

meets

y -

axis and having an arbitrary tangent of

C_{2}

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

12 e

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Let tangent drawn to the parabola C1:y2=4ax at P meets the y-axis at Q. Another parabola C2 with vertex Q and focus (0,0) is drawn which passes through (2a,0). Identify which of the following statements can be CORRECT?

Let tangent drawn to the parabola $C_{1} : y^{2} = 4 a x$ at $P$ meets the $y$ -axis at $Q$ . Another parabola $C_{2}$ with vertex $Q$ and focus $(0, 0)$ is drawn which passes through $(2 a, 0) .$ Identify which of the following statements can be CORRECT?