Intersection of Tangents at Two Points

Intersection of Tangents at Two Points

For circle:

$$x^2+y^2=a^2$$

the tangents at parameters $$\alpha$$ and $$\beta$$ are:

$$x\cos\alpha+y\sin\alpha=a$$

$$x\cos\beta+y\sin\beta=a$$

Their intersection is:

$$\left(a\cos\frac{\alpha+\beta}{2}\sec\frac{\alpha-\beta}{2},a\sin\frac{\alpha+\beta}{2}\sec\frac{\alpha-\beta}{2}\right)$$

Usage

- Used to find the intersection point of two tangents when their points of contact are given parametrically.