Let $P: y^{2}=4ax, a>0$ be a parabola with focus $S$. Let the tangents to the parabola $P$ make an angle of $\frac{\pi}{4}$ 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 collinear is:

JEE Main · 2022 · 29 Jun 2022 · Shift 2 · Parabola

Let

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

be a parabola with focus

S

. Let the tangents to the parabola

P

make an angle of

\frac{π}{4}

with the line

y = 3 x + 5

touch the parabola

P

A

and

B

. Then the value of

a

for which

A, B

and

S

are collinear is:

Answer: (D)

The line

y = 3 x + 5

has slope 3. Lines making angle

\frac{π}{4}

with this line have slopes

m

satisfying

\frac{m - 3}{1 + 3 m} = tan \frac{π}{4} = 1

. Solving:

m - 3 = \pm (1 + 3 m)

gives

m = - 2

m = \frac{1}{2}

. These two slopes are perpendicular (product

= - 1

). For a parabola, if two tangents are perpendicular, they intersect on the directrix. Since the focus

S

lies on the axis of the parabola and the tangents at

A

and

B

are perpendicular, the chord

A B

passes through the focus for any value of

a > 0

. Therefore

A, S, B

are collinear for any

a > 0

JEE Main · 2022 · 29 Jun 2022 · Shift 2 · Single correct option