Question

Two tangents are drawn from a point to $y^2=4ax$ if these are normals to $x^2=4by$ then

• A. $a^2>b^2$
• B. $a^2>4b^2$
• C. $a^2>8b^2$
• D. $4a^2>b^2$

Answer 1

Answer: (C)

$a^2>8b^2$

Let the slopes of both lines from the external given point be $m_1$ & $m_2$

Equations of tangents to $y^2=4ax$ are

$\Rightarrow y=m_{1}x+\left(\frac{a}{m_1}\right)$ .....$e{q}^n\left(1\right)$

& $\Rightarrow y=m_{2}x+\left(\frac{a}{m_2}\right)$

Similarly equation of normal to $x^2=4by$ are

$\Rightarrow y=m_{1}x+2b+\left(\frac{b}{m_1^2}\right)$ .....$e{q}^n\left(2\right)$

$\Rightarrow y=m_{2}x+2b+\left(\frac{b}{m_2^2}\right)$

Thus, since lines written in both equations (1) & (2) are same

$\left(\frac{a}{m_1}\right)=2b+\frac{b}{m_1^2}$

$\Rightarrow$ $2b{m}_1^2-a{m}_1+b=0$

Now, the discriminant is positive

i.e. $D>0$

$\Rightarrow$ $(-a{)}^2-4\times (2b)\times (b)>0$

$\Rightarrow$ $a^2>8b^2$

Hence, option (C) is correct.