Question

If the circles $x^2 + y^2 + 2ax + cy + a - 0$ and $x^2 + y^2 - 3ax + dy - 1 = 0$ intersect in two distinct points P and Q, then the line $5x + by - a - 0$ passes through P and Q for:

• A. exactly two values of a
• B. infinitely many values of a
• C. no value of a
• D. exactly one value of a

Answer 1

Answer: (C)

If $S_1 = 0$ and $S_2 = 0$ are two circles, then the equation of line passes through the points of intersection of $S_1 = 0$ and $S_2 = 0$, is $S_1 - S_2 = 0$.
Equation of circles are
$S_1 = x^2 + y^2 + 2ax + cy + a = 0$
and $S_2 = x^2 + y^2 - 3ax + dy -1 = 0$
respectively.
Chord through intersection points P and Q of the circles $S_1 = 0$ and $S_2 = 0$ is $S_1 - S_2 = 0$.
$\therefore \quad5ax + \left(c - d\right)y + a + 1 = 0$
On comparing it with $5x + by - a = 0$, we get
$\frac{5a}{5} = \frac{c-d}{b} = \frac{a+1}{-a}$
$\Rightarrow \quad a\left(-a\right) = a + 1 \Rightarrow a^{2} + a+ 1= 0$
Which gives no real value of a.
$\therefore $ The line will passes through P and Q for no value of a