88828
If one of the lines represented by \(6 x^{2}+k x y+y^{2}=0\) is \(2 x+y=0\), then what is the value of \(k\) ?
1 3
2 4
3 5
4 -5
Explanation:
(C) : Pair of straight line passing through origin-
\(6 x^{2}+k x y+y^{2}=0 \tag{i}\)
One line \(y=-2 x\) which must satisfies equation (i)
\(\therefore 6 \mathrm{x}^{2}+\mathrm{kx}(-2 \mathrm{x})+(-2 \mathrm{x})^{2}=0\)
\(\Rightarrow \mathrm{x}^{2}(10-2 \mathrm{k})=0\)
\(\Rightarrow \mathrm{K}=5\)
SCRA-2012
Straight Line
88829
The equation \(\sqrt{(x-2)^{2}+y^{2}}+\sqrt{(x+2)^{2}+y^{2}}=4\) represents
1 a circle
2 a parabola
3 a pair of lines
4 an ellipse
Explanation:
(C) : We have
\(\sqrt{(x-2)^{2}+y^{2}}+\sqrt{(x+2)^{2}+y^{2}}=4\)
\(\Rightarrow \quad \sqrt{x^{2}+4-4 x+y^{2}}+\sqrt{x^{2}+4+4 x+y^{2}}=4\)
By hit and trial method
\(\mathrm{x}= \pm 2, \mathrm{y}=0\)
\(\mathrm{x}= \pm 1, \mathrm{y}=0\)
\(\mathrm{x}=0 . \mathrm{v}=0\)
Which represent a pair of straight line.
AMU-2015
Straight Line
88795
If \(\lambda x^{2}-10 x y+12 y^{2}+5 x-16 y-3=0\), represents a pair of straight lines, then the value of \(\lambda\) is
1 4
2 3
3 2
4 1
Explanation:
(C) : Given, \(\lambda x^{2}-10 x y+12 y^{2}+5 x-16 y-3=0\)
This will represent a pair of straight lines if
\(a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0\)
\(\Rightarrow \lambda(12)(-3)+2(-8)\left(\frac{5}{2}\right)(-5)-\lambda(-8)^{2}-12\left(\frac{5}{2}\right)^{2}\) \(+3(-5)^{2}=0\)
\(\Rightarrow-36 \lambda+200-64 \lambda-75+75=0\)
\(\Rightarrow 100 \lambda=200 \Rightarrow \lambda=2\)
SRM JEE-2011
Straight Line
88796
The separate equations of the lines represented by \(4 x^{2}-y^{2}+2 x+y=0\) are
1 \(2 x-y+1=0,2 x-y=0\)
2 \(2 x-2 y+1=0, x+2 y=0\)
3 \(2 x-y=0,2 x+y+1=0\)
4 \(2 x-y+1=0,2 x+y=0\)
Explanation:
(D) : The equation of line is,
\(4 \mathrm{x}^{2}-\mathrm{y}^{2}+2 \mathrm{x}+\mathrm{y}=0\)
\((2 \mathrm{x}+\mathrm{y})(2 \mathrm{x}-\mathrm{y})+(2 \mathrm{x}+\mathrm{y})=0\)
\((2 \mathrm{x}+\mathrm{y})(2 \mathrm{x}-\mathrm{y}+1)=0\)
\(2 \mathrm{x}+\mathrm{y}=0\) and \(2 \mathrm{x}-\mathrm{y}+1=0\)
88828
If one of the lines represented by \(6 x^{2}+k x y+y^{2}=0\) is \(2 x+y=0\), then what is the value of \(k\) ?
1 3
2 4
3 5
4 -5
Explanation:
(C) : Pair of straight line passing through origin-
\(6 x^{2}+k x y+y^{2}=0 \tag{i}\)
One line \(y=-2 x\) which must satisfies equation (i)
\(\therefore 6 \mathrm{x}^{2}+\mathrm{kx}(-2 \mathrm{x})+(-2 \mathrm{x})^{2}=0\)
\(\Rightarrow \mathrm{x}^{2}(10-2 \mathrm{k})=0\)
\(\Rightarrow \mathrm{K}=5\)
SCRA-2012
Straight Line
88829
The equation \(\sqrt{(x-2)^{2}+y^{2}}+\sqrt{(x+2)^{2}+y^{2}}=4\) represents
1 a circle
2 a parabola
3 a pair of lines
4 an ellipse
Explanation:
(C) : We have
\(\sqrt{(x-2)^{2}+y^{2}}+\sqrt{(x+2)^{2}+y^{2}}=4\)
\(\Rightarrow \quad \sqrt{x^{2}+4-4 x+y^{2}}+\sqrt{x^{2}+4+4 x+y^{2}}=4\)
By hit and trial method
\(\mathrm{x}= \pm 2, \mathrm{y}=0\)
\(\mathrm{x}= \pm 1, \mathrm{y}=0\)
\(\mathrm{x}=0 . \mathrm{v}=0\)
Which represent a pair of straight line.
AMU-2015
Straight Line
88795
If \(\lambda x^{2}-10 x y+12 y^{2}+5 x-16 y-3=0\), represents a pair of straight lines, then the value of \(\lambda\) is
1 4
2 3
3 2
4 1
Explanation:
(C) : Given, \(\lambda x^{2}-10 x y+12 y^{2}+5 x-16 y-3=0\)
This will represent a pair of straight lines if
\(a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0\)
\(\Rightarrow \lambda(12)(-3)+2(-8)\left(\frac{5}{2}\right)(-5)-\lambda(-8)^{2}-12\left(\frac{5}{2}\right)^{2}\) \(+3(-5)^{2}=0\)
\(\Rightarrow-36 \lambda+200-64 \lambda-75+75=0\)
\(\Rightarrow 100 \lambda=200 \Rightarrow \lambda=2\)
SRM JEE-2011
Straight Line
88796
The separate equations of the lines represented by \(4 x^{2}-y^{2}+2 x+y=0\) are
1 \(2 x-y+1=0,2 x-y=0\)
2 \(2 x-2 y+1=0, x+2 y=0\)
3 \(2 x-y=0,2 x+y+1=0\)
4 \(2 x-y+1=0,2 x+y=0\)
Explanation:
(D) : The equation of line is,
\(4 \mathrm{x}^{2}-\mathrm{y}^{2}+2 \mathrm{x}+\mathrm{y}=0\)
\((2 \mathrm{x}+\mathrm{y})(2 \mathrm{x}-\mathrm{y})+(2 \mathrm{x}+\mathrm{y})=0\)
\((2 \mathrm{x}+\mathrm{y})(2 \mathrm{x}-\mathrm{y}+1)=0\)
\(2 \mathrm{x}+\mathrm{y}=0\) and \(2 \mathrm{x}-\mathrm{y}+1=0\)
88828
If one of the lines represented by \(6 x^{2}+k x y+y^{2}=0\) is \(2 x+y=0\), then what is the value of \(k\) ?
1 3
2 4
3 5
4 -5
Explanation:
(C) : Pair of straight line passing through origin-
\(6 x^{2}+k x y+y^{2}=0 \tag{i}\)
One line \(y=-2 x\) which must satisfies equation (i)
\(\therefore 6 \mathrm{x}^{2}+\mathrm{kx}(-2 \mathrm{x})+(-2 \mathrm{x})^{2}=0\)
\(\Rightarrow \mathrm{x}^{2}(10-2 \mathrm{k})=0\)
\(\Rightarrow \mathrm{K}=5\)
SCRA-2012
Straight Line
88829
The equation \(\sqrt{(x-2)^{2}+y^{2}}+\sqrt{(x+2)^{2}+y^{2}}=4\) represents
1 a circle
2 a parabola
3 a pair of lines
4 an ellipse
Explanation:
(C) : We have
\(\sqrt{(x-2)^{2}+y^{2}}+\sqrt{(x+2)^{2}+y^{2}}=4\)
\(\Rightarrow \quad \sqrt{x^{2}+4-4 x+y^{2}}+\sqrt{x^{2}+4+4 x+y^{2}}=4\)
By hit and trial method
\(\mathrm{x}= \pm 2, \mathrm{y}=0\)
\(\mathrm{x}= \pm 1, \mathrm{y}=0\)
\(\mathrm{x}=0 . \mathrm{v}=0\)
Which represent a pair of straight line.
AMU-2015
Straight Line
88795
If \(\lambda x^{2}-10 x y+12 y^{2}+5 x-16 y-3=0\), represents a pair of straight lines, then the value of \(\lambda\) is
1 4
2 3
3 2
4 1
Explanation:
(C) : Given, \(\lambda x^{2}-10 x y+12 y^{2}+5 x-16 y-3=0\)
This will represent a pair of straight lines if
\(a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0\)
\(\Rightarrow \lambda(12)(-3)+2(-8)\left(\frac{5}{2}\right)(-5)-\lambda(-8)^{2}-12\left(\frac{5}{2}\right)^{2}\) \(+3(-5)^{2}=0\)
\(\Rightarrow-36 \lambda+200-64 \lambda-75+75=0\)
\(\Rightarrow 100 \lambda=200 \Rightarrow \lambda=2\)
SRM JEE-2011
Straight Line
88796
The separate equations of the lines represented by \(4 x^{2}-y^{2}+2 x+y=0\) are
1 \(2 x-y+1=0,2 x-y=0\)
2 \(2 x-2 y+1=0, x+2 y=0\)
3 \(2 x-y=0,2 x+y+1=0\)
4 \(2 x-y+1=0,2 x+y=0\)
Explanation:
(D) : The equation of line is,
\(4 \mathrm{x}^{2}-\mathrm{y}^{2}+2 \mathrm{x}+\mathrm{y}=0\)
\((2 \mathrm{x}+\mathrm{y})(2 \mathrm{x}-\mathrm{y})+(2 \mathrm{x}+\mathrm{y})=0\)
\((2 \mathrm{x}+\mathrm{y})(2 \mathrm{x}-\mathrm{y}+1)=0\)
\(2 \mathrm{x}+\mathrm{y}=0\) and \(2 \mathrm{x}-\mathrm{y}+1=0\)
88828
If one of the lines represented by \(6 x^{2}+k x y+y^{2}=0\) is \(2 x+y=0\), then what is the value of \(k\) ?
1 3
2 4
3 5
4 -5
Explanation:
(C) : Pair of straight line passing through origin-
\(6 x^{2}+k x y+y^{2}=0 \tag{i}\)
One line \(y=-2 x\) which must satisfies equation (i)
\(\therefore 6 \mathrm{x}^{2}+\mathrm{kx}(-2 \mathrm{x})+(-2 \mathrm{x})^{2}=0\)
\(\Rightarrow \mathrm{x}^{2}(10-2 \mathrm{k})=0\)
\(\Rightarrow \mathrm{K}=5\)
SCRA-2012
Straight Line
88829
The equation \(\sqrt{(x-2)^{2}+y^{2}}+\sqrt{(x+2)^{2}+y^{2}}=4\) represents
1 a circle
2 a parabola
3 a pair of lines
4 an ellipse
Explanation:
(C) : We have
\(\sqrt{(x-2)^{2}+y^{2}}+\sqrt{(x+2)^{2}+y^{2}}=4\)
\(\Rightarrow \quad \sqrt{x^{2}+4-4 x+y^{2}}+\sqrt{x^{2}+4+4 x+y^{2}}=4\)
By hit and trial method
\(\mathrm{x}= \pm 2, \mathrm{y}=0\)
\(\mathrm{x}= \pm 1, \mathrm{y}=0\)
\(\mathrm{x}=0 . \mathrm{v}=0\)
Which represent a pair of straight line.
AMU-2015
Straight Line
88795
If \(\lambda x^{2}-10 x y+12 y^{2}+5 x-16 y-3=0\), represents a pair of straight lines, then the value of \(\lambda\) is
1 4
2 3
3 2
4 1
Explanation:
(C) : Given, \(\lambda x^{2}-10 x y+12 y^{2}+5 x-16 y-3=0\)
This will represent a pair of straight lines if
\(a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0\)
\(\Rightarrow \lambda(12)(-3)+2(-8)\left(\frac{5}{2}\right)(-5)-\lambda(-8)^{2}-12\left(\frac{5}{2}\right)^{2}\) \(+3(-5)^{2}=0\)
\(\Rightarrow-36 \lambda+200-64 \lambda-75+75=0\)
\(\Rightarrow 100 \lambda=200 \Rightarrow \lambda=2\)
SRM JEE-2011
Straight Line
88796
The separate equations of the lines represented by \(4 x^{2}-y^{2}+2 x+y=0\) are
1 \(2 x-y+1=0,2 x-y=0\)
2 \(2 x-2 y+1=0, x+2 y=0\)
3 \(2 x-y=0,2 x+y+1=0\)
4 \(2 x-y+1=0,2 x+y=0\)
Explanation:
(D) : The equation of line is,
\(4 \mathrm{x}^{2}-\mathrm{y}^{2}+2 \mathrm{x}+\mathrm{y}=0\)
\((2 \mathrm{x}+\mathrm{y})(2 \mathrm{x}-\mathrm{y})+(2 \mathrm{x}+\mathrm{y})=0\)
\((2 \mathrm{x}+\mathrm{y})(2 \mathrm{x}-\mathrm{y}+1)=0\)
\(2 \mathrm{x}+\mathrm{y}=0\) and \(2 \mathrm{x}-\mathrm{y}+1=0\)
88828
If one of the lines represented by \(6 x^{2}+k x y+y^{2}=0\) is \(2 x+y=0\), then what is the value of \(k\) ?
1 3
2 4
3 5
4 -5
Explanation:
(C) : Pair of straight line passing through origin-
\(6 x^{2}+k x y+y^{2}=0 \tag{i}\)
One line \(y=-2 x\) which must satisfies equation (i)
\(\therefore 6 \mathrm{x}^{2}+\mathrm{kx}(-2 \mathrm{x})+(-2 \mathrm{x})^{2}=0\)
\(\Rightarrow \mathrm{x}^{2}(10-2 \mathrm{k})=0\)
\(\Rightarrow \mathrm{K}=5\)
SCRA-2012
Straight Line
88829
The equation \(\sqrt{(x-2)^{2}+y^{2}}+\sqrt{(x+2)^{2}+y^{2}}=4\) represents
1 a circle
2 a parabola
3 a pair of lines
4 an ellipse
Explanation:
(C) : We have
\(\sqrt{(x-2)^{2}+y^{2}}+\sqrt{(x+2)^{2}+y^{2}}=4\)
\(\Rightarrow \quad \sqrt{x^{2}+4-4 x+y^{2}}+\sqrt{x^{2}+4+4 x+y^{2}}=4\)
By hit and trial method
\(\mathrm{x}= \pm 2, \mathrm{y}=0\)
\(\mathrm{x}= \pm 1, \mathrm{y}=0\)
\(\mathrm{x}=0 . \mathrm{v}=0\)
Which represent a pair of straight line.
AMU-2015
Straight Line
88795
If \(\lambda x^{2}-10 x y+12 y^{2}+5 x-16 y-3=0\), represents a pair of straight lines, then the value of \(\lambda\) is
1 4
2 3
3 2
4 1
Explanation:
(C) : Given, \(\lambda x^{2}-10 x y+12 y^{2}+5 x-16 y-3=0\)
This will represent a pair of straight lines if
\(a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0\)
\(\Rightarrow \lambda(12)(-3)+2(-8)\left(\frac{5}{2}\right)(-5)-\lambda(-8)^{2}-12\left(\frac{5}{2}\right)^{2}\) \(+3(-5)^{2}=0\)
\(\Rightarrow-36 \lambda+200-64 \lambda-75+75=0\)
\(\Rightarrow 100 \lambda=200 \Rightarrow \lambda=2\)
SRM JEE-2011
Straight Line
88796
The separate equations of the lines represented by \(4 x^{2}-y^{2}+2 x+y=0\) are
1 \(2 x-y+1=0,2 x-y=0\)
2 \(2 x-2 y+1=0, x+2 y=0\)
3 \(2 x-y=0,2 x+y+1=0\)
4 \(2 x-y+1=0,2 x+y=0\)
Explanation:
(D) : The equation of line is,
\(4 \mathrm{x}^{2}-\mathrm{y}^{2}+2 \mathrm{x}+\mathrm{y}=0\)
\((2 \mathrm{x}+\mathrm{y})(2 \mathrm{x}-\mathrm{y})+(2 \mathrm{x}+\mathrm{y})=0\)
\((2 \mathrm{x}+\mathrm{y})(2 \mathrm{x}-\mathrm{y}+1)=0\)
\(2 \mathrm{x}+\mathrm{y}=0\) and \(2 \mathrm{x}-\mathrm{y}+1=0\)
