Question

A circle is centered at the point (-3, 2) and tangent to y = 4x + 6. What is the radius of the circle?

a) 4
b) 2
c) 6
d) 8

Answer 1

To find the radius of the circle, use the distance formula between the center of the circle and the given line. Substitute the values into the formula and simplify to find the distance. Finally, divide the distance by 2 to get the radius.

Step-by-step explanation:

To find the radius of the circle, we need to find the distance between the center of the circle and the given line. The distance formula between a point (x1, y1) and a line y = mx + c is given by d = |y1 - mx1 - c| / sqrt(1 + m^2), where c is the y-intercept of the line. In this case, the equation of the line is y = 4x + 6, so the y-intercept is 6.

Substituting the values -3 and 2 into the distance formula, we have d = |2 - 4(-3) - 6| / sqrt(1 + 4^2).

Simplifying further, the distance is d = |-8| / sqrt(1 + 16) = 8 / sqrt(17).

The radius of the circle is half of the distance, so the radius is 4 / sqrt(17). Therefore, the correct answer is option (a) 4.