Question

If the coordinates of the point of contact of the circles \( x^2 + y^2 - 4x + 8y + 4 = 0 \) and \( x^2 + y^2 + 2x = 0 \) is \( (a, b) \), then \( a + 2b \) is:

• A. \( -1 \)
• B. \( -2 \)
• C. \( 0 \)
• D. \( 1 \)

Hint:

When solving for the point of contact between two circles, complete the square to obtain the circle's centers and radii. Then solve the system of equations to find the coordinates.

Answer 1

The Correct Option is B

The problem involves finding the shortest distance between the point \( P(-2, -4) \) and the circle \( S \). The shortest distance from a point to a circle is along the line connecting the point to the center of the circle, which is the perpendicular distance.
To solve for \( PQ \), we can use the formula for the distance from a point to a circle, knowing that the center and radius of the circle can be derived from the equation of the circumcircle.
The distance formula for \( PQ \) gives \( 5 - \sqrt{5} \) as the shortest distance. Therefore, the correct answer is \( 5 - \sqrt{5} \).