Find the equation of common tangent of the circle $x^2+y^2-8x=0$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$

Question

Equation the condition of tangency of both the conics $$4m\pm \sqrt {9m^2-4}=4\sqrt {1+m^2}$$my problem is easy. Are there any tips to reduce such heavy calculations?

Answer 1

The equation of any tangent of the hyperbola at $(3\sec t,2\tan t)$ will be $$x(3\sec t)/9-y(2\tan t)/4=1\iff 2x\tan t -3y\tan t-6=0$$

Now if this has to be a tangent of the given circle, the distance from the center will be same

$$4=\dfrac{|2\sec t(4)+0-6|}{\sqrt{(2\sec t)^2+(3\tan t)^2}}$$

Take square in both sides and use $$\tan ^2t=\sec^2t-1$$ to form a quadratic equation in $\sec t$