Question

The absolute difference between the squares of the radii of the two circles passing through the point \( (-9, 4) \) and touching the lines \( x + y = 3 \) and \( x - y = 3 \), is equal to:

Hint:

For problems involving two tangential circles and points of intersection, use the distance formula between the center and line to find the radius.

Answer 1

Correct Answer: 768

Let the center of the first circle be \( (a, 0) \), with radius \( r_1 \). The equation of the circle is: \[ (x - a)^2 + y^2 = r_1^2 \] Now, the distance from the center of the circle to the line \( x + y = 3 \) is the radius \( r_1 \). The distance formula for a point to a line \( Ax + By + C = 0 \) is: \[ \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] Substituting the values, we find the relationship between \( a \) and \( r_1 \). Similarly, for the second circle, we use the equation of the second line \( x - y = 3 \). The result of the calculations is the absolute difference between the squares of the radii: \[ |r_1^2 - r_2^2| = 768 \]