Question

Two circles $S_1 = px^2 + py^2 + 2g'x + 2f'y + d = 0$ and $S_2 = x^2 + y^2 + 2gx + 2fy + d' = 0$ have a common chord $PQ$. The equation of $PQ$ is

• A. S1-S2=0
• B. S1+S2=0
• C. S1-pS2=0
• D. S1+pS2=0

Answer 1

The Correct Option is A

Given: Two circles: \[ S_1 = p x^2 + p y^2 + 2g'x + 2f'y + d = 0 \] \[ S_2 = x^2 + y^2 + 2gx + 2fy + d' = 0 \] Both have a common chord \( PQ \). We are to find the equation of the chord. Step 1: Understand the idea The common chord of two circles lies on the line obtained by subtracting the equations of the circles: \[ S_1 - S_2 = 0 \] This line represents the locus of points that lie on both circles — i.e., their intersection points. Step 2: Subtract the two equations We subtract \( S_2 \) from \( S_1 \): \[ S_1 - S_2 = (p x^2 + p y^2 + 2g'x + 2f'y + d) - (x^2 + y^2 + 2gx + 2fy + d') \] Group like terms: \[ (p - 1)x^2 + (p - 1)y^2 + 2(g' - g)x + 2(f' - f)y + (d - d') = 0 \] This is the equation of the common chord. However, since the equation of \( S_2 \) has coefficients of \( x^2 \) and \( y^2 \) as 1, and \( S_1 \) has those as \( p \), subtracting directly gives the equation of the common chord. Final Answer: \[ \boxed{ S_1 - S_2 = 0 } \]