Find the tangents to circle

Question

Let $ \Gamma : x^{2} + y^{2} - 6x - 4y + 8 = 0 $ be a circle.

Find the equations of the tangents to $ \Gamma $ which pass through $ D(8, 7) $.

Answer 1

HINT:

The equation of any straight line passing through $D(8,7)$ can be written as $$\dfrac{y-7}{x-8}=m\iff y=mx+7-8m$$

Replace this value of $y$ in the equation of the circle to form a Quadratic Equation in $x,$ whose roots represents the abscissa of the intersection

For tangency, both root should be same. This will give the two possible values of $m$