Circles
Circle
Allen Star Batch
Grade 11
Question:
From a point $A(2, 2)$ two chords $AB$ and $AC$ of $1$ unit length are drawn to the circle $x^2 + y^2 = 8$. If the equation of the chord $BC$ is given by $ax + by = 15$, then the value of $a + b$ is __________.
Step-by-Step Solution
Key Concept: The radical axis (difference of two circle equations) gives the equation of the common chord.
The circle with center $(2, 2)$ and radius $1$ has equation $S_1 = (x-2)^2 + (y-2)^2 = 1$. The chord $AB$ is found by setting $S_1 - S_2 = 0$, which gives the radical axis equation $4x + 4y = 15$. This represents the common chord of intersection between the two circles.
Correct Answer: 8