Question

If the points $\left( a v ^ { 2 } , 2 a v \right)$are the extremities of a focal chord of the parabola $y ^ { 2 } = 4 a x$then

• A. $u v - 1 = 0$
• B. $u v + 1 = 0$
• C. $u + v = 0$
• D. $u - v = 0$

Answer 1

Answer: (B)

$u v + 1 = 0$

Answer 2

Equation of focal chord for the parabola $\left( a v ^ { 2 } , 2 a v \right)$

⇒ $y - 2 a u = \frac { 2 a v - 2 a u } { a v ^ { 2 } - a u ^ { 2 } } \left( x - a u ^ { 2 } \right)$

⇒

⇒ $y - 2 a u = \frac { 2 } { v + u } \left( x - a u ^ { 2 } \right)$

It this is focal chord, so it would passes through focus (a, 0)

⇒ $- u v - u ^ { 2 } = 1 - u ^ { 2 }$, ∴ $u v + 1 = 0$

Trick : Given points $t _ { 1 } t _ { 2 } = - 1$. Hence $u v + 1 = 0$