Conic-Sections Question 544
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
Options:
A) Exactly one value of a
B) No value of a
C) Infinitely many values of a
D) Exactly two values of a
Show Answer
Answer:
Correct Answer: B
Solution:
[b] $ S_1=x^{2}+y^{2}+2ax+cy+a=0 $ $ S_2=x^{2}+y^{2}-3ax+dy-1=0 $ Equation of common chord of circles $ S_1 $ and $ S_2 $ is given by $ S_1-S_2=0 $
$ \Rightarrow 5ax+(c-d)y+a+1=0 $ Given that $ 5x+by-a=0 $ passes through P and Q
$ \therefore $ The two equation should represent the same line
$ \Rightarrow \frac{a}{1}=\frac{c-d}{b}=\frac{a+1}{-a}\Rightarrow a+1=-a^{2} $
$ \Rightarrow a^{2}+a+1=0 $ No real value of a.
BETA
