Question

The equations of the normal at the ends of the latus rectum of the parabola $y ^ { 2 } = 4 a x$are given by

• A. $x ^ { 2 } - y ^ { 2 } - 6 a x + 9 a ^ { 2 } = 0$
• B. $x ^ { 2 } - y ^ { 2 } - 6 a x - 6 a y + 9 a ^ { 2 } = 0$
• C. $x ^ { 2 } - y ^ { 2 } - 6 a y + 9 a ^ { 2 } = 0$
• D. None of these

Answer 1

Answer: (A)

$x ^ { 2 } - y ^ { 2 } - 6 a x + 9 a ^ { 2 } = 0$

Answer 2

The coordinates of the ends of the latus rectum of the parabola $y ^ { 2 } = 4 a x$are $( a , 2 a )$ and $( a , - 2 a )$ respectively.

The equation of the normal at $( a , 2 a )$ to $y ^ { 2 } = 4 a x$is $y - 2 a = \frac { - 2 a } { 2 a } ( x - a )$

Or $x + y - 3 a = 0$ .....(i)

Similarly the equation of the normal at (a, –2a) is

$x - y - 3 a = 0$ .....(ii)

The combined equation of (i) and (ii) is

$x ^ { 2 } - y ^ { 2 } - 6 a x + 9 a ^ { 2 } = 0$