Pair of lines problem The Next CEO of Stack OverflowDetermining the type of conic and drawing a graph of itHyperbola is a pair of straight lines?Is there a plane that passes through a pair of lines that have no points in common?Area of a triangle - straight linesArea of triangle bounded by line and degenerate “crossed lines” conicAngle between a pair of tangent linesPoint circle from pair of straight line equationEquation of pair of perpendicular lines in inclined axes?Pair of straight lines problemArea bounded by three non concurrent straight linesEquation of a pair of straight lines through the origin and passing through the intersection of two curves
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Pair of lines problem
The Next CEO of Stack OverflowDetermining the type of conic and drawing a graph of itHyperbola is a pair of straight lines?Is there a plane that passes through a pair of lines that have no points in common?Area of a triangle - straight linesArea of triangle bounded by line and degenerate “crossed lines” conicAngle between a pair of tangent linesPoint circle from pair of straight line equationEquation of pair of perpendicular lines in inclined axes?Pair of straight lines problemArea bounded by three non concurrent straight linesEquation of a pair of straight lines through the origin and passing through the intersection of two curves
$begingroup$
If the pair of straight lines $x^2+2xy+ay^2$ & $ax^2+2xy+y^2$ have exactly one line in common, then the combined equation of the two lines is given by
A. $3x^2+8xy-3y^2$
B. $3x^2+10xy+3y^2$
C. $-3x^2+2xy+y^2$
D. $x^2+2xy-3y^2$
My approach
Let the equation of lines are
lx+my=0(Common line)
px+qy=0..Line 1
rx+ty=0..Line 2
Equation of line 1 and common line
$lpx^2+(lq+mp)xy+mqy^2=0$
Equation of line 2 and common line
$rlx^2+(lt+rm)xy+tmy^2=0$
Not able to compare it with the question.
geometry
asked Mar 20 at 9:10
Samar Imam ZaidiSamar Imam Zaidi
1,6041520
$endgroup$
add a comment |
$begingroup$
If the pair of straight lines $x^2+2xy+ay^2$ & $ax^2+2xy+y^2$ have exactly one line in common, then the combined equation of the two lines is given by
A. $3x^2+8xy-3y^2$
B. $3x^2+10xy+3y^2$
C. $-3x^2+2xy+y^2$
D. $x^2+2xy-3y^2$
My approach
Let the equation of lines are
lx+my=0(Common line)
px+qy=0..Line 1
rx+ty=0..Line 2
Equation of line 1 and common line
$lpx^2+(lq+mp)xy+mqy^2=0$
Equation of line 2 and common line
$rlx^2+(lt+rm)xy+tmy^2=0$
Not able to compare it with the question.
geometry
asked Mar 20 at 9:10
Samar Imam ZaidiSamar Imam Zaidi
1,6041520
$endgroup$
$begingroup$
I don't understand ... Shouldn't the equation of a line be an equation with an $=$-sign? How is just the number $x^2 + 2xy + ay^2$ a straight line?
$endgroup$
– Matti P.
Mar 20 at 9:15
$begingroup$
@ Matti P. see math.stackexchange.com/questions/2311197
$endgroup$
– cgiovanardi
Mar 20 at 19:27
add a comment |
$begingroup$
If the pair of straight lines $x^2+2xy+ay^2$ & $ax^2+2xy+y^2$ have exactly one line in common, then the combined equation of the two lines is given by
A. $3x^2+8xy-3y^2$
B. $3x^2+10xy+3y^2$
C. $-3x^2+2xy+y^2$
D. $x^2+2xy-3y^2$
My approach
Let the equation of lines are
lx+my=0(Common line)
px+qy=0..Line 1
rx+ty=0..Line 2
Equation of line 1 and common line
$lpx^2+(lq+mp)xy+mqy^2=0$
Equation of line 2 and common line
$rlx^2+(lt+rm)xy+tmy^2=0$
Not able to compare it with the question.
geometry
asked Mar 20 at 9:10
Samar Imam ZaidiSamar Imam Zaidi
1,6041520
$endgroup$
If the pair of straight lines $x^2+2xy+ay^2$ & $ax^2+2xy+y^2$ have exactly one line in common, then the combined equation of the two lines is given by
A. $3x^2+8xy-3y^2$
B. $3x^2+10xy+3y^2$
C. $-3x^2+2xy+y^2$
D. $x^2+2xy-3y^2$
My approach
Let the equation of lines are
lx+my=0(Common line)
px+qy=0..Line 1
rx+ty=0..Line 2
Equation of line 1 and common line
$lpx^2+(lq+mp)xy+mqy^2=0$
Equation of line 2 and common line
$rlx^2+(lt+rm)xy+tmy^2=0$
Not able to compare it with the question.
geometry
geometry
asked Mar 20 at 9:10
Samar Imam ZaidiSamar Imam Zaidi
1,6041520
asked Mar 20 at 9:10
Samar Imam ZaidiSamar Imam Zaidi
1,6041520
asked Mar 20 at 9:10
Samar Imam ZaidiSamar Imam Zaidi
1,6041520
asked Mar 20 at 9:10
Samar Imam ZaidiSamar Imam Zaidi
1,6041520
asked Mar 20 at 9:10
Samar Imam ZaidiSamar Imam Zaidi
1,6041520
1,6041520
$begingroup$
I don't understand ... Shouldn't the equation of a line be an equation with an $=$-sign? How is just the number $x^2 + 2xy + ay^2$ a straight line?
$endgroup$
– Matti P.
Mar 20 at 9:15
$begingroup$
@ Matti P. see math.stackexchange.com/questions/2311197
$endgroup$
– cgiovanardi
Mar 20 at 19:27
add a comment |
$begingroup$
I don't understand ... Shouldn't the equation of a line be an equation with an $=$-sign? How is just the number $x^2 + 2xy + ay^2$ a straight line?
$endgroup$
– Matti P.
Mar 20 at 9:15
$begingroup$
@ Matti P. see math.stackexchange.com/questions/2311197
$endgroup$
– cgiovanardi
Mar 20 at 19:27
$begingroup$
I don't understand ... Shouldn't the equation of a line be an equation with an $=$-sign? How is just the number $x^2 + 2xy + ay^2$ a straight line?
$endgroup$
– Matti P.
Mar 20 at 9:15
$begingroup$
I don't understand ... Shouldn't the equation of a line be an equation with an $=$-sign? How is just the number $x^2 + 2xy + ay^2$ a straight line?
$endgroup$
– Matti P.
Mar 20 at 9:15
$begingroup$
@ Matti P. see math.stackexchange.com/questions/2311197
$endgroup$
– cgiovanardi
Mar 20 at 19:27
$begingroup$
@ Matti P. see math.stackexchange.com/questions/2311197
$endgroup$
– cgiovanardi
Mar 20 at 19:27
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Hint.
If $x^2+2 x y + a y^2$ and $a x^2+2 x y + y^2$ have a common linear factor, then
$$
x^2+2 x y + a y^2-(a x^2+2 x y + y^2) = (1-a) x^2 -(1-a) y^2 = (1-a)(x^2-y^2) = (1-a)(x+y)(x-y)
$$
has this same factor.
answered Mar 20 at 9:42
CesareoCesareo
9,5473517
$endgroup$
add a comment |
$begingroup$
Hint:
If $x=p$ is the common root
$$p^2+2py+ay^2=0$$ and $$ap^2+2py+y^2=0$$
Solve for $p,p^2$
Use $p^2=(p)^2$ to eliminate $p$
answered Mar 20 at 9:19
lab bhattacharjeelab bhattacharjee
228k15158279
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint.
If $x^2+2 x y + a y^2$ and $a x^2+2 x y + y^2$ have a common linear factor, then
$$
x^2+2 x y + a y^2-(a x^2+2 x y + y^2) = (1-a) x^2 -(1-a) y^2 = (1-a)(x^2-y^2) = (1-a)(x+y)(x-y)
$$
has this same factor.
answered Mar 20 at 9:42
CesareoCesareo
9,5473517
$endgroup$
add a comment |
$begingroup$
Hint.
If $x^2+2 x y + a y^2$ and $a x^2+2 x y + y^2$ have a common linear factor, then
$$
x^2+2 x y + a y^2-(a x^2+2 x y + y^2) = (1-a) x^2 -(1-a) y^2 = (1-a)(x^2-y^2) = (1-a)(x+y)(x-y)
$$
has this same factor.
answered Mar 20 at 9:42
CesareoCesareo
9,5473517
$endgroup$
add a comment |
$begingroup$
Hint.
If $x^2+2 x y + a y^2$ and $a x^2+2 x y + y^2$ have a common linear factor, then
$$
x^2+2 x y + a y^2-(a x^2+2 x y + y^2) = (1-a) x^2 -(1-a) y^2 = (1-a)(x^2-y^2) = (1-a)(x+y)(x-y)
$$
has this same factor.
answered Mar 20 at 9:42
CesareoCesareo
9,5473517
$endgroup$
Hint.
If $x^2+2 x y + a y^2$ and $a x^2+2 x y + y^2$ have a common linear factor, then
$$
x^2+2 x y + a y^2-(a x^2+2 x y + y^2) = (1-a) x^2 -(1-a) y^2 = (1-a)(x^2-y^2) = (1-a)(x+y)(x-y)
$$
has this same factor.
answered Mar 20 at 9:42
CesareoCesareo
9,5473517
answered Mar 20 at 9:42
CesareoCesareo
9,5473517
answered Mar 20 at 9:42
CesareoCesareo
9,5473517
answered Mar 20 at 9:42
CesareoCesareo
9,5473517
9,5473517
add a comment |
add a comment |
$begingroup$
Hint:
If $x=p$ is the common root
$$p^2+2py+ay^2=0$$ and $$ap^2+2py+y^2=0$$
Solve for $p,p^2$
Use $p^2=(p)^2$ to eliminate $p$
answered Mar 20 at 9:19
lab bhattacharjeelab bhattacharjee
228k15158279
$endgroup$
add a comment |
$begingroup$
Hint:
If $x=p$ is the common root
$$p^2+2py+ay^2=0$$ and $$ap^2+2py+y^2=0$$
Solve for $p,p^2$
Use $p^2=(p)^2$ to eliminate $p$
answered Mar 20 at 9:19
lab bhattacharjeelab bhattacharjee
228k15158279
$endgroup$
add a comment |
$begingroup$
Hint:
If $x=p$ is the common root
$$p^2+2py+ay^2=0$$ and $$ap^2+2py+y^2=0$$
Solve for $p,p^2$
Use $p^2=(p)^2$ to eliminate $p$
answered Mar 20 at 9:19
lab bhattacharjeelab bhattacharjee
228k15158279
$endgroup$
Hint:
If $x=p$ is the common root
$$p^2+2py+ay^2=0$$ and $$ap^2+2py+y^2=0$$
Solve for $p,p^2$
Use $p^2=(p)^2$ to eliminate $p$
answered Mar 20 at 9:19
lab bhattacharjeelab bhattacharjee
228k15158279
answered Mar 20 at 9:19
lab bhattacharjeelab bhattacharjee
228k15158279
answered Mar 20 at 9:19
lab bhattacharjeelab bhattacharjee
228k15158279
answered Mar 20 at 9:19
lab bhattacharjeelab bhattacharjee
228k15158279
228k15158279
add a comment |
add a comment |
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$begingroup$
I don't understand ... Shouldn't the equation of a line be an equation with an $=$-sign? How is just the number $x^2 + 2xy + ay^2$ a straight line?
$endgroup$
– Matti P.
Mar 20 at 9:15
$begingroup$
@ Matti P. see math.stackexchange.com/questions/2311197
$endgroup$
– cgiovanardi
Mar 20 at 19:27