Multiple choice

Consider a family of circles passing through two fixed points $A(3, 7)$ and $B(6, 5)$. The chord, in which the circle $x^2 + y^2 - 4x - 6y- 3 = 0$ cuts each member of the family of circles, passes through a fixed point $(a, b)$. Find the numerical value of $a + 3b$.

  1. 3

  2. 13

  3. 25

  4. 35
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The family of circles passing through A and B is S + kL = 0. The common chord of any member and the fixed circle is the radical axis. The radical axis of the family and the fixed circle passes through the intersection of the fixed circle and the line AB. Solving for the fixed point yields the result.

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