Question

The condition so that the line $lx+my+n=0$ may touch the parabola $y^2=8x$

• A. $m^2=8ln$
• B. $m^2=2ln$
• C. $8m^2=ln$
• D. $2m^2=ln$

Answer 1

Answer: (D)

Given equation of the line is $lx+my+n=0$
$\Rightarrow my=-lx-n$
$\Rightarrow y=\frac{-l}{m}x-\frac{n}{m}$
On comparing with $y=Mx+C$, we get
$M=\frac{-l}{m}$ and $C=\frac{-n}{m}$
Again, given equation of the parabola is $y^2=8x$
On comparing with $y^2=4ax$, we get
$4a=8\Rightarrow a=2$
Now, by condition of tangency,
$C=\frac{a}{M}$
$\Rightarrow \frac{-n}{m}=\frac{2}{-1/m}$ [from Eq.(i)]
$\Rightarrow \frac{n}{m}=\frac{2m}{l}\Rightarrow \ln =2m^2$