Ella Moon

2022-02-11

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

ashuhra6e

Beginner2022-02-12Added 15 answers

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

Next, find$\frac{dy}{dx}$ 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 $\frac{dy}{dx}=-\frac{x}{y}$ .

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

$\frac{dy}{dx}=-\frac{4}{3}.$

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

Next, find

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

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