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 π 4 with the line y = 3 x + 5 touch the parabola P at A and B . Then the value of a for which A ,   B and S are collinear is:

Question

Let P : y 2 = 4 a x , &#160; a &#62; 0 be a parabola with focus S .Let the tangents to the parabola P make an angle of &#960; 4 with the line y = 3 x + 5 touch the parabola P at A and B . Then the value of a for which A , &#160; B and S are collinear is:

Aicharya · Accepted Answer

Lines making angle $\frac{\pi}{4}$ with $y=3x+5$ So using formula $\tan \alpha = \frac{m_2 - m_1}{1 + m_1 m_2}$, we have slope $-2 \& \frac{1}{2}$.

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 = 4 a x ,   a > 0 be a parabola with focus S .Let the tangents to the parabola P make…

Solution

Let P : y 2 = 4 a x , &#160; a &#62; 0 be a parabola with focus S .Let the tangents to the parabola P make…

Solution

Let P : y 2 = 4 a x , a > 0 be a parabola with focus S .Let the tangents to the parabola P make…