CPM Homework Help : PC Problem 8-60

Find the equation of the line tangent to the circle $x^{2} + y^{2} = 25$ at the point $\left(3, 4\right)$ . (Recall that a tangent line to a circle is perpendicular to that circle’s radius at the point of tangency.)

Step 1:

Draw a diagram of the situation.

Step 2:

Find the slope of the radius, AP.

Step 3:

Find the slope of the tangent line.

It is the negative reciprocal of the slope of the radius found above.

Step 4:

Now you have the point of tangency and the slope. Write the equation using the point-slope form.

Circle with center labeled A, point in first quadrant, labeled, P = (3, comma 4), dashed segment tangent to circle through the point, segment from, A to P, is hypotenuse of right tirangle, whose vertical leg labeled 4, & horizontal leg labeled 3.

$\frac{\text{rise}}{\text{run}}=\frac{4}{3}$

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