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A variable line L is drawn through O (0,...

A variable line L is drawn through O (0,0) to meet the lines L 1 ​ :y−x−10=0 and L 2 ​ :y−x−20=0 at the points A and B respectively. A point P is taken on L such that OP 2 ​ = OA 1 ​ + OB 1 ​ and P,A,B lies on same side of origin O. The locus of P is

A

3x+3y=40

B

3x+3y+40 =0

C

3x-3y=40

D

3y-3x=40

Text Solution

Verified by Experts

The correct Answer is:
D
Let the parametric equation of the line drawn be
`(x)/("cos" theta) = (y)/("sin" theta) =r`
`"or " x = r "cos" theta, y = r"sin" theta`
Putting it in `L_(1)`, we get
`r "sin" theta, = r"cos" theta +10`
`"or "(1)/(OA) = ("sin" theta-"cos" theta)/(10)`
Similarly, putting the general point of drawn line in the equation of `L_(2)` , we get
`(1)/(OB) = ("sin" theta- "cos" theta)/(20)`
`"Let "P-=(h,k) " and " OP =r." Then, "r "cos"theta = h, r " sin"theta = k. " We have "`
`(2)/(r) = ("sin" theta-"cos" theta)/(10) + ("sin" theta-"cos" theta)/(20)`
`"or " 40 = 3r "sin" theta-3r "cos" theta`
or 3y-3x=40
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