What will be the equation of tangent

Question

How to prove that $3x-4y+11=0$ is a tangent to the circle $x^2+y^2-8y+15=0$? What will be the equation of the other tangent which is parallel to the line $3x=4y$?

Answer 1

The centre of circle is $(0,4)$ and radius is $1$.
So the distance of the the line from the centre of circle must be 1, which it is:

$$\left|\dfrac{0-16+11}{\sqrt{3^2 + 4^2}}\right| = 1$$

Let the other tangent be $3x-4y+k=0$. Again use the result of distance of line from centre of circle.

$$\left|\dfrac{0-16+k}{\sqrt{3^2 + 4^2}}\right| = 1$$

Using this, you get two values of $k$: $11$ and $21$