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

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 $y=-\frac{4}{3}\left(x-4\right)+3$, which can also be written as $y=-\frac{4}{3}x+\frac{25}{3}$.

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