MHT CET · Maths · Pair of Lines
If the equation \(3 x^{2}+10 x y+3 y^{2}+16 y+k=0\) represents a pair of lines, then
the value of \(k\) is
- A \(-21\)
- B 21
- C 12
- D \(-12\)
Answer & Solution
Correct Answer
(D) \(-12\)
Step-by-step Solution
Detailed explanation
(A)
Comparing the given equation with \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0\), we get, \(a=3, h=5, b=3, g=0, f=8, c=k\).
Now, given equation represents a pair of lines.
\(\therefore a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0\)
\(\therefore\) (3) (3) \((\mathrm{k})+2(8)(0)(5)-3(8)^{2}-3(0)^{2}-\mathrm{k}(5)^{2}=0\)
\(\therefore 9 \mathrm{k}+0-192-0-25 \mathrm{k}=0 \Rightarrow 16 \mathrm{k}=-192 \Rightarrow \mathrm{k}=-12\)
Comparing the given equation with \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0\), we get, \(a=3, h=5, b=3, g=0, f=8, c=k\).
Now, given equation represents a pair of lines.
\(\therefore a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0\)
\(\therefore\) (3) (3) \((\mathrm{k})+2(8)(0)(5)-3(8)^{2}-3(0)^{2}-\mathrm{k}(5)^{2}=0\)
\(\therefore 9 \mathrm{k}+0-192-0-25 \mathrm{k}=0 \Rightarrow 16 \mathrm{k}=-192 \Rightarrow \mathrm{k}=-12\)
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