Question

How do i create a equation for common tangent to anyb2 circles

• A. The problem is ill-posed as it does not provide specific equations for the two circles.
• B. The equation of a common tangent can be found by considering the distance from the centers of the circles to the tangent line.
• C. Common tangents can only exist if the circles do not intersect.
• D. The equation of a common tangent is always a horizontal line.

Answer 1

Answer: (B)

The equation of a common tangent can be found by considering the distance from the centers of the circles to the tangent line.

Answer 2

To find the equation of a common tangent to two circles, we use the property that the perpendicular distance from the center of each circle to the tangent line must be equal to the radius of that circle. If the circles are $C_1: (x - h_1)^2 + (y - k_1)^2 = r_1^2$ and $C_2: (x - h_2)^2 + (y - k_2)^2 = r_2^2$, and the tangent line is $ax + by + c = 0$, then the conditions are: $\frac{|ah_1 + bk_1 + c|}{\sqrt{a^2 + b^2}} = r_1$ $\frac{|ah_2 + bk_2 + c|}{\sqrt{a^2 + b^2}} = r_2$ These equations, along with the relationship derived from them, allow us to determine the coefficients $a, b, c$ and thus the equation(s) of the common tangent(s).