Let P: y ^ 2 = 4ax, a > 0 be a parabola with focus S .Let the tangents to the parabola P make an angle of (π)/(4) with the line y = 3x + 5 touch the parabola P at A and B . Then the value of a for which A, B and S are

JEE Main 202229 JuneEvening Shift

Exam: JEE Main
Year: 2022
Date: 29 June
Shift: Evening Shift
Subject: Mathematics
Chapter: Parabola

JEE Main 2022 · 29 June · Evening Shift · Mathematics · Parabola

Let P: y ^ 2 = 4ax, a > 0 be a parabola with focus S .Let the...

Let

P: {y}^{2} = 4ax, a > 0

be a parabola with focus

S

.Let the tangents to the parabola

P

make an angle of

\dfrac{\pi}{4}

with the line

y = 3x + 5

touch the parabola

P

A

and

B

. Then the value of

a

for which

A, B

and

S

are collinear is: Let P: y ^ 2 = 4ax, a > 0 be a parabola with focus S .Let the tangents to the parabola P make an angle of (π)/(4) with

Options

A $8$ only
B $2$ only
C $\dfrac{1}{4}$ only
Dany $a > 0$

Correct answer

D. any $a > 0$

Step-by-step solution

Lines making angle $\dfrac{\pi}{4}$ with $y = 3x + 5$

So using formula $tan\alpha = \dfrac{{m}_{2} - {m}_{1}}{1 + {m}_{1}{m}_{2}}$ , we have slope $- 2 \text{ and } \dfrac{1}{2}$ .

Which are perpendicular to each-other so, $A, S, B$ are collinear for all $a > 0$ . (by property that perpendicular tangents will lie on directrix and chord of contact will pass through focus, so the points are collinear )

Let P: y ^ 2 = 4ax, a > 0 be a parabola with focus S .Let the tangents to the parabola P make an angle of (π)/(4) with

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