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:
Show Hint
For problems involving two tangential circles and points of intersection, use the distance formula between the center and line to find the radius.
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: \[ |r_1^2 - r_2^2| = 768 \]
