Question

Let each of the equations \(x^{2}+2 x y+a y^{2}=0 \& a x^{2}+2 x y+y^{2}=0\) represent two straight lines passing through the origin. If they have a common line, then the other two lines are given by
(A) \(x-y=0, x-3 y=0\)
(B) \(x+3 y=0,3 x+y=0\)
(C) \(3 x+y=0,3 x-y=0\) (D) \((3 x-2 y)=0, x+y=0\)

Answer 1

Ans: (B)
Hint: \(\left(\frac{x}{y}\right)^{2}+2\left(\frac{x}{y}\right)+a=0 \& a\left(\frac{x}{y}\right)^{2}+2\left(\frac{x}{y}\right)+1=0\) have exactly one root in common (taking \(\frac{x}{y}\) as a single variable).
By, \(\left(a_{1} b_{2}-a_{2} b_{1}\right)\left(b_{1} c_{2}-b_{2} c_{1}\right)=\left(a_{1} c_{2}-a_{2} c_{1}\right)^{2}\)
We get : \(\Rightarrow a=1\) or \(-3\)
a cannot be 1
Taking \(a=-3\), roots of 1 st equation : \(1,-3\) and 2 nd equation : \(1,-\frac{1}{3}\)
So other lines : \(\frac{x}{y}=-3\) and \(\frac{x}{y}=-\frac{1}{3}\)

Answer 2

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Answer 3

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