Question

At what point(s) on the circle x 2 + y 2 = 13 is its tangent line parallel to the line 3x − 2y = 6.

Answer 1

$$3x − 2y = 6$$

$$y =\dfrac{3}{2}x-3$$

$$slope=m=\dfrac{3}{2}$$

$$x^ 2 + y^ 2 = 13$$

Differentiate both sides with respect to $x$ and use the Chain Rule

$$2x+2yy'=0$$

$$x+yy'=0$$

Then

$$x_1+y_1y'(x_1)=0$$

Substitute

$$y'(x_1)=slope=m=\dfrac{3}{2}$$

$$x_1+\dfrac{3}{2}y_1=0$$

$$x_1=-\dfrac{3}{2}y_1$$

Point $(x_1, y_1)$ lies on the circle

$$x_1^2+y_1^2=13$$

Then

$$(-\dfrac{3}{2}y_1)^2+y_1^2=13$$

$$y_1^2=4$$

$$y_1=\pm2$$

Point $(-3, 2)$ and Point $(3, -2).$