Question

If a length of focal chord of the parabola $y^2=4ax$ at a distance of b from the vertex 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 B $a^3=b^{2}c$

Parabola: $y^2=4ax......\left(i\right)$

Let the end points of focal chord be $({at}^2,2at)4(\frac{a}{t^2},\frac{-2a}{t})$

Let the chord be $=>y=mx+c^{\prime }......\left(ii\right)$

It must pass through $(a,0)=>ma+c^{\prime }=0$

$=>c^{\prime }=-ma$

Now equation $\left(ii\right)$ becomes$=>y=mx-ma......\left(iii\right)$

perpendicular distance from origin to the line$\left(iii\right)=>b=\left|\frac{0-0+ma}{\sqrt{1+m^2}}\right|$

$=>b=\left(\sqrt{1+m^2}\right)=ma$

$=>b^2+b^2{m}^2=m^2{a}^2$

$=>m^2(a^2-b^2)=b^2$

$=>m=\pm \sqrt{\frac{b^2}{(a^2-b^2)}}$

$=>y=\sqrt{\frac{b^2}{a^2-b^2}}(x-a)$

Length of chord $=>c=\frac{1}{m}\sqrt{a(1+m^2)(a-m{c}^{\prime })}$

$=>c=\frac{\frac{1}{b^2}}{a^2-b^2}\sqrt{a(1+\frac{b^2}{a^2-b^2})(a+(\sqrt{\frac{b^2}{a^2-b^2}\left)\right(\sqrt{\frac{b^2}{a^2-b^2}\left)\right(a)}}}$

$=>c=\frac{a^2-b^2}{b^2}\sqrt{\frac{a\left(a^2\right)}{a^2-b^2}\left(a\right)\left(\frac{a^2}{a^2-b^2}\right)+1}$

$=>c=\frac{a^2-b^2}{b^2}.\frac{a^3}{a^2-b^2}$

$=>b^{2}c=a^3$.