Question

The locus of the midpoint of the line segment joining the focus to a moving point on the parabola $y ^ { 2 } = 4 a x$is another parabola with the directrix

• A. $x = - a$
• B. $x = - \frac { a } { 2 }$
• C. $x = 0$
• D. $x = \frac { a } { 2 }$

Answer 1

Answer: (C)

$x = 0$

Answer 2

Let $M ( \alpha , \beta )$be the mid point of PS.

$\alpha = \frac { a t ^ { 2 } + a } { 2 } , \beta = \frac { 2 a t + 0 } { 2 }$ ⇒

∴ $2 a \alpha = \beta ^ { 2 } + a ^ { 2 }$

∴ The locus is $y ^ { 2 } = \frac { 4 a } { 2 } \left( x - \frac { a } { 2 } \right) = 4 b ( x - b ) , \left\{ b = \frac { a } { 2 } \right\}$

∴ Directrix is $( x - b ) + b = 0$or $x = 0$