Question

If $Ax+By=1$ is a normal to the curve $ay=x^2$ , then

• A. $4A^2(1-aB)=a{B}^3$
• B. $4A^2(2+aB)=a{B}^3$
• C. $4A^2(1+aB)+a{B}^3=0$
• D. $2A^2(2-aB)=a{B}^3$

Answer 1

The correct option is D $2A^2(2-aB)=a{B}^3$

$Ax+By=1$
$\Rightarrow y=-\frac{A}{B}x+\frac{1}{B}\cdots \left(i\right)$
This is a normal to the parabola $x^2=ay$
We know that, the equation of normal to parabola $x^2=4by$ is
$y=mx+2b+\frac{b}{m^2}$
For the given curve $b=\frac{a}{4}$
Then, equation of normal is
$y=mx+\frac{a}{2}+\frac{a}{4m^2}\cdots \left(ii\right)$

Comparing $\left(ii\right)$ with the given line $\left(i\right)$, we get
$m=-\frac{A}{B}\frac{1}{B}=\frac{a}{2}+\frac{a}{4m^2}\Rightarrow \frac{1}{B}-\frac{a}{2}=\frac{a{B}^2}{4A^2}\therefore (2-aB)2A^2=a{B}^3$