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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