Question

The length of a focal chord of the parabola $y^2=4ax$ at a distance $b$ from the vertex is $c$, then

• A. $2a^2=bc$
• B. $a^3=b^{2}c$
• C. $ac=b^2$
• D. $b^{2}c=4a^3$

Answer 1

The correct option is D $b^{2}c=4a^3$

Let one end of the focal chord be $(a{t}^2,2at)$, then the length of the that chord is
$c=a{(t+\frac{1}{t})}^2\cdots \left(1\right)$
Now equation of the focal chord will be
$y-0=\left(\frac{2at-0}{a{t}^2-a}\right)(x-a)$
As we know that vertex coordinates of the parabola is $(0,0)$
Hence perpendicular distance from origin to the chord will be
$b=\frac{\left|\frac{-2at}{t^2-1}\right|}{\sqrt{1+\frac{4t^2}{(t^2-1{)}^2}}}=\frac{|2at|}{t^2+1}\cdots \left(2\right)$
from $\left(1\right)\times (2{)}^2$, $t$ will get eliminated and we get
$b^{2}c=4a^3$