CircleHard
Question
If the circles x2 + y2 + 2ax + cy + a = 0 and x2 + y2 - 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
Options
A.exactly one value of a
B.no value of a
C.infinitely many values of a
D.exactly two values of a
Solution
S1 = x2 + y2 + 2ax + cy + a = 0
S2 = x2 + y2 - 3ax + dy - 1 = 0
Equation of radical axis of S1 and S2
S1 - S2 = 0
⇒ 5ax + (c - d)y + a + 1 = 0
Given that 5x + by - a = 0 passes through P and Q
⇒
⇒ a + 1 = - a2
a2 + a + 1 = 0
No real value of a
S2 = x2 + y2 - 3ax + dy - 1 = 0
Equation of radical axis of S1 and S2
S1 - S2 = 0
⇒ 5ax + (c - d)y + a + 1 = 0
Given that 5x + by - a = 0 passes through P and Q
⇒
⇒ a + 1 = - a2
a2 + a + 1 = 0
No real value of a
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