Test Index

Parabola

Section: Mathematics

© examsnet.com

Question : 5 of 31

Marks: +1, -0

Consider the parabola

y^2=4x

S

be the focus of the parabola. A pair of tangents drawn to the parabola from the point

P=(-2,1)

meet the parabola at

P_1

P_2

Q_1

Q_2

be points on the lines

S P_1

S P_2

respectively such that

P Q_1

is perpendicular to

S P_1

P Q_2

is perpendicular to

{SP}_2

. Then, which of the following is/are TRUE ?

[JEE Adv 2022 P1]

$S Q_1=2$
$Q_1 Q_2 = \;\frac{3\sqrt{10}}{5}$
${PQ}_1=3$
${SQ}_2=1$

Solution:

Let equation of tangent with slope '

m

' be

T: y=mx+\;\frac{1}{m}

T

: passes through

(-2,1)

so

1=-2 m+\;\frac{1}{m}

\Rightarrow m=-1

or

m=\;\frac{1}{2}

Points are given by

\left(\;\frac{a}{m^2},\;\frac{2a}{m}\right)

So, one point will be

(1,-2)\&(4,4)

Let

P_1(4,4)\& P_2(1,-2)

P_1 S: 4x-3y-4=0

P_2 S: x-1=0

{PQ}_1 = \left| \frac{4(-2)-3(1)-4}{5} \right| = 3

SP = \sqrt{10} ; {PQ}_2 = 3 ; {SQ}_1 = 1 = {SQ}_2

\;\frac{1}{2}\left(\frac{Q_1 Q_2}{2}\right)\times\sqrt{10}=\frac{1}{2}\times 3\times 1\;\;

(comparing Areas)

\Rightarrow Q_1 Q_2 = \;\frac{2\times 3}{\sqrt{10}} = \;\frac{3\sqrt{10}}{5}

© examsnet.com

Go to Question:

Prev Question

Next Question