Coordinate Geometry
Common chord of two circles; condition for a line to pass through intersection points
Grade Class 12
Question:
If the circles $x^2+y^2+10\alpha x+\beta y+\alpha=0$ and $x^2+y^2-5\alpha x+\gamma y-1=0$ intersect in two distinct points $A$ and $B$, then the line $15x+\delta y-\alpha=0$ passes through $A$ and $B$ for
infinitely many values of $\alpha$
exactly two values of $\alpha$
exactly one value of $\alpha$
no value of $\alpha$
Step-by-Step Solution
Key Concept: Subtract the two circle equations to get the equation of the common chord AB. Compare with the given line $15x+\delta y-\alpha=0$ by matching coefficients.
Common chord: $15\alpha x+(\beta+\gamma)y+(\alpha+1)=0$. Comparing with $15x+\delta y-\alpha=0$: $15\alpha/15 = (\alpha+1)/(-\alpha)$ gives $\alpha^2+\alpha+1=0$, which has no real roots. Hence no value of $\alpha$.
Correct Answer: 4