Question

How do you determine of the tangent line at the point (4,3) that lies on the circle `x^2 + y^2 = 25`?

Answer 1

First, confirm that the point `(4,3)` is on the circle: `4^2+3^2=16+9=25`.

Next, find `dy/dx` by implicitly assuming `y` is a function of `x`, using the Chain Rule, and then doing some algebra: `2x+2y\cdot \frac{dy}{dx}=0` so that `dy/dx=-x/y`.

The slope of the tangent line to the circle at the point `(x,y)=(4,3)` is therefore `dy/dx=-4/3`.

This means the equation of the tangent line to the circle at that point can be written as `y=-4/3(x-4)+3`, which can also be written as `y=-4/3x+25/3`.