Two perpendicular tangents to y^2 = 4ax always intersect on the l… - Vidyadip

MCQ

Two perpendicular tangents to ${y^2} = 4ax$ always intersect on the line, if

A
$x = a$
✓
$x + a = 0$
C
$x + 2a = 0$
D
$x + 4a = 0$

✓

Answer

Correct option: B.

$x + a = 0$

b
(b) We know that tangent to the parabola at points ${t_1}$ and ${t_2}$ are ${t_1}y = x + at_1^2$ and ${t_2}y = x + at_2^2.$

Since tangents are perpendicular to the parabola,

therefore, $\frac{1}{{{t_1}}}.\frac{1}{{{t_2}}} = - 1$ or ${t_1}{t_2} = - 1$.

We also know that their point of intersection $ = (a{t_1}{t_2},\,a({t_1} + {t_2}))$ $ = \,( - a,\,a({t_1} + {t_2})).$

Thus these points lie on directrix $x = - \,a$ or $x + a = 0$.

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