Question

I. Common tangent to the parabolas $y^2=32x$ and $x^2=108y$ is $2x+3y+36=0$
II: Common tangent to the parabolas $y^2=32x$ and $x^2=-108y$ is $2x-3y+36=0$.

Which of the above statements is correct?

• A. only I
• B. only II
• C. Both I and II
• D. neither I nor II

Answer 1

Answer: (C)

Both I and II

I. $y^2=32x$ and $x^2=108y$

The tangent to $y^2=32x$ is $y=mx+\frac{8}{m}$

and tangent to $x^2=108y$ is $y=mx-27m^2$

Comparing the slopes of the two equations of the tangents we get

$\frac{8}{m}=-27m^2\Rightarrow m^3=-\frac{8}{27}\Rightarrow m=\frac{-2}{3}$

Thus putting the value of $m$ in the equation of tangent we get $3y+2x+36=0$

II. $y^2=32x$ and $x^2=-108y$

The tangent to $y^2=32x$ is $y=mx+\frac{8}{m}$

and tangent to $x^2=-108y$ is $y=mx+27m^2$

Comparing the slopes of the two equations of the tangents we get

$\frac{8}{m}=27m^2\Rightarrow m^3=\frac{8}{27}\Rightarrow m=\frac{2}{3}$

Thus putting the value of $m$ in the equation of tangent we get $-3y+2x+36=0$