Let P : y2 = 4ax, a > 0 be a parabola with focus S.Let the tangents to the parabola P make an angle of π4with 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

Lines making angleπ4with y = 3x + 5tanπ4=m-31+3m⇒1+3m=±(m-3) 1+3m=m-3⇒m=-2 1+3m=-m+2⇒m=12Which are perpendicular to each-other so, A, S, Bare collinear for all a > 0.

Let P : y2 = 4ax, a > 0 be a parabola with focus S.Let the tangents to the parabola P make an angle of π4with 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

Lines making angleπ4with y = 3x + 5tanπ4=m-31+3m⇒1+3m=±(m-3) 1+3m=m-3⇒m=-2 1+3m=-m+2⇒m=12Which are perpendicular to each-other so, A, S, Bare collinear for all a > 0.

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Let P : y² = 4ax, 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 = 3x + 5 touch the parabola P at A and B. Then the value of a for which A,B and S are collinear is

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8 only

2 only

$\frac{1}{4}$ only

any a > 0

answer is D.

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Detailed Solution

Lines making angle $\frac{π}{4}$ with y = 3x + 5
$\tan \frac{π}{4} = |\frac{m - 3}{1 + 3 m}| \Rightarrow 1 + 3 m = \pm (m - 3) 1 + 3 m = m - 3 \Rightarrow m = - 2 1 + 3 m = - m + 2 \Rightarrow m = \frac{1}{2}$
Which are perpendicular to each-other so, A, S, B
are collinear for all a > 0.