Question

Length of the focal chords of the parabola $y^2=4ax$ at a distance p from the vertex is

• A. $\frac{2a^2}{p}$
• B. $\frac{a^2}{p^2}$
• C. $\frac{4a^3}{p^2}$
• D. $\frac{p^2}{a}$

Answer 1

The correct option is C $\frac{4a^3}{p^2}$

$y^2=4ax$

Slope of $OP=$ Slope of $OQ$

$\Rightarrow t_2=\frac{-1}{t_1}$

$\therefore$ $P({at}^2,2at)$ & $Q(\frac{a}{t^2},\frac{-2a}{t})$

Let length of focal chord be $C$.

$\therefore$ $\sqrt{{({at}^2-\frac{a}{t^2})}^2+{(2at+\frac{2a}{t})}^2}=C$

$\Rightarrow \sqrt{a^2{(t^2-\frac{1}{t^2})}^2+{\left(2a\right)}^2{(t+\frac{1}{t})}^2}=C$

$\Rightarrow \sqrt{a^2{(t+\frac{1}{t})}^2{(t-\frac{1}{t})}^2+{\left(2a\right)}^2{(t+\frac{1}{t})}^2}=C$

$\Rightarrow \sqrt{a^2{(t+\frac{1}{t})}^2[{(t-\frac{1}{t})}^2+4]}=C$

$\Rightarrow \sqrt{a^2{(t+\frac{1}{t})}^2(t+\frac{1}{t^2}-2+4)}=C$

$\Rightarrow \sqrt{a^2{(t+\frac{1}{t})}^2{(t+\frac{1}{t})}^2}=C$

$\Rightarrow a{(t+\frac{1}{t})}^2=C⟶\left(1\right)$

Equation of $PQ$

$y+\frac{2a}{t}=\frac{2at+2a/t}{{at}^2-\frac{a}{t^2}}(x-\frac{a}{t^2})$

$\Rightarrow y+\frac{2a}{t}=\frac{\frac{{2t}^2+2}{t}}{\frac{t^4-1}{t^2}}(x-\frac{a}{t^2})$

$\Rightarrow y(t^4-1)+\frac{2a}{t}(t^4-1)=2(t^2+1)t(x-\frac{a}{t^2})$

$\Rightarrow y(t^2+1)(t^2-1)+\frac{2a}{t}(t^2+1)(t^2-1)=2t(t^2+1)(x-\frac{a}{t^2})$

$\Rightarrow y(t^2-1)+\frac{2a}{t}(t^2-1)=2t(x-\frac{a}{t^2})$

$\Rightarrow y(t^2-1)+\frac{2a}{t}(t^2-1)=2tx-\frac{2at}{t^2}$

$\Rightarrow 2tx-y(t^2-1)-\frac{2at}{t^2}-\frac{{2at}^2}{t}+\frac{2a}{t}=0$

$\Rightarrow 2tx-y(t^2-1)-2at=0$

$\therefore$ Equation of focal chord is $2tx-y(t^2-1)-2at=0$

$Q=\frac{|A_1{x}_1+B_1{y}_1+C_1|}{\sqrt{A_1^2+B_1^2}}$ $(0,0)$

$\therefore$ $P=\left|\frac{-2at}{\sqrt{{\left(2t\right)}^2+{(t^2-1)}^2}}\right|=\frac{2at}{\sqrt{{4t}^2+t^4+1-{2t}^2}}$

$=\frac{2at}{\sqrt{t^4+1+{2t}^2}}=\frac{2at}{\sqrt{{(t^2+1)}^2}}=\frac{2at}{t^2+1}$

$\Rightarrow \frac{t^2+1}{t}=\frac{2a}{p}⟶\left(2\right)$

$\Rightarrow t+\frac{1}{t}=\frac{2a}{P}$

$\therefore$ From $\left(1\right)$, we get,

$a{\left(\frac{2a}{P}\right)}^2=C$

$\Rightarrow C=\frac{{4a}^2}{P^2}.a$

$\Rightarrow C=\frac{{4a}^3}{P^2}$

$\therefore$ Length of focal chord is $\frac{{4a}^3}{P^2}$.