How do you determine of the tangent line at the point (4,3) that lies on the circle x2+y2=25?

Question

ashuhra6e · Accepted Answer

First, confirm that the point (4,3) is on the circle:

4^{2} + 3^{2} = 16 + 9 = 25

.
Next, find

\frac{d y}{d x}

by implicitly assuming y is a function of x, using the Chain Rule, and then doing some algebra:

2 x + 2 y \cdot \frac{d y}{d x} = 0

so that

\frac{d y}{d x} = - \frac{x}{y}

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

\frac{d y}{d x} = - \frac{4}{3} .

This means the equation of the tangent line to the circle at that point can be written as

y = - \frac{4}{3} (x - 4) + 3

, which can also be written as

y = - \frac{4}{3} x + \frac{25}{3}

.

How do you determine of the tangent line at the