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Find the joint equation of the pair of lines through the origin , one of which is parallel to 2x+y=5 and other is perpendicular to 3x 4x+7=0.
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The combined equation of the pair of lines through origin such that one is parallel to 3x + 2y = 3 and the other is perpendicular to 6x + 3y + 17 = 0 is ______.
3x2 + 4xy + 4y2 = 0
3x2 - 4xy - 4y2 = 0
3x2 - 4xy + 4y2 = 0
3x2 - 8xy + 4y2 = 0
The combined equation of the pair of lines through origin such that one is parallel to 3x + 2y = 3 and the other is perpendicular to 6x + 3y + 17 = 0 is 3x2 - 4xy - 4y2 = 0.
Explanation:
Slope ofline 3x + 2y = 3 is `(-3)/2`
∴ Line parallel to 3x + 2y = 3 and passing through origin is y = `(-3)/2`x
⇒ 3x + 2y = 0
Slope of 6x + 3y + 17 = 0 is - 2
∴ Line perpendicular to 6x + 3y + 17 = 0 and passing through origin is y = `1/2`x
⇒ x - 2y = 0
Their combined equation is
(3x + 2y) (x - 2y) = 0
⇒ 3x2 - 6xy + 2yx - 4y2 = 0
⇒ 3x2 - 4xy - 4y2 = 0
Concept: Formation of Joint Equation and Separation of Equations from a Given Equation
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Find the joint equation of the line passing through the origin and perpendicular to the lines x + 2y = 19 and 3x + y = 18
Let L1 and L2 be the lines passing through the origin and perpendicular to the lines x + 2y = 19 and 3x + y = 18 respectively.
Slopes of the lines x + 2y = 19 and 3x + y = 18 are `-1/2` and `- 3/1` = -3 respectively.
∴ slopes of the lines L1 and L2 are 2 and `1/3` respectively.
Since the lines L1 and L2 pass through the origin, their equations are
y = 2x and y = `1/3`x
i.e. 2x - y = 0 and x - 3y = 0
∴ their combined equation is
(2x - y)(x - 3y) = 0
∴ 2x2 - 6xy - xy + 3y2 = 0
∴ 2x2 - 7xy + 3y2 = 0
Concept: Combined Equation of a Pair Lines
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