Find the tangent equation to the circle

Question

The circle is given as

$$x^2+y^2+z^2-7y+2z-8= 0$$ $$3x-2y+4z+3=0$$

at the point $(-3,5,4)$.

I know the answer will be in the form of

$$\frac{(x+3)}{l}=\frac{( y -5 )}{m}=\frac{( z-4)}{n}$$

but how to find $<l,m,n>$?

This will be to perpendicular to normal of given plane need one more relation.

Answer 1

Hints:

What is the desired leading vector? Indeed, if you give some considereations, you'lll find that it is the vector product of the Gradient vectors of both surfaces: $$\nabla f_{(-3,5,4)}\times\nabla g_{(-3,5,4)}=\vec{u}$$

Answer 2

Hint: Fine the center of circle Find the point(s) where the tangent(s) meet(s) the circle Form the equations