Question

Number of common tangent to the curves $xy=c^2$ & $y^2=4ax$ is:

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is B $1$

Equation of the tangent with slope $m$ to the parabola $y^2=4ax$ is

$y=mx+\frac{a}{m}$ ...................... (1)

This line is also tangent to the hyperbola

$xy=c^2$ ................... (2)

We can find the point of contact on the hyperbola by substituting $y=mx+\frac{a}{m}$ in equation (2),

$x(mx+\frac{a}{m})=c^2$

$m^2{x}^2+ax-m{c}^2=0$

There is one point of contact, so the above equation will have two equal roots.

Discriminant $=0$

$a^2+4m^3{c}^2=0$

$m^3=-\frac{a^2}{4c^2}$

$m=-{\left(\frac{a}{2c}\right)}^{\frac{2}{3}}$ ................... (3)

There is only one value of the slope $m$, which is negative.

So, there will be only one common tangent.