Tangents are drawn at the point of intersections of the circles $x^2 + y^2 = 1$ and $x^2 + y^2 - (\lambda + 6)x + (8 - 2\lambda)y - 3 = 0$. ($\lambda$ being the variable). Then the locus of the point of intersection of these tangents is:
Step-by-Step Solution
Key Concept: The radical axis is the locus of points with equal power with respect to both circles, found by subtracting circle equations.
The radical axis of two intersecting circles is found by subtracting their equations: $S_1 - S_2 = 0$, giving $hx + yk = 1$. Comparing with the line equation $(\lambda + 6)x + (2\lambda - 8)y + 2 = 0$, we equate coefficients. Solving the system of ratios yields $2x - y + 10 = 0$, or equivalently $2x - y = -10$.
Correct Answer: 1