Question

# A variable line is drawn through the intersection point of the lines 3x+4y−12=0 and x+2y−5=0 meeting the coordinate axes at the points A and B. Locus of mid point of segment AB is

A
4x+3y=4xy
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B
3x+4y=3xy
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C
3x+4y=4xy
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D
none
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Solution

## The correct option is B 3x+4y=4xyREF.ImageIntegration of (3x+4y−12=0&x+2y=5)x=5−2y3x+4y=12⇒3(5−2y)+4y=12⇒15−6y+4y=12⇒2y=3⇒y=3/2x=2Intersection pt. →(2,3/2)eqn. of line : y=mx+c32=2m+c⇒c=32−2my=mx+cx=0→y=c⇒k=cy=mx+cy=0→x=−c/m⇒h=−c/mMid pt. of (h,0)&(0,k) is (h/2,k/2)(h2,k2)−(−c2m,c2)=(2m−3/22m,3/2−2m2)=(1−34m,34−m)=(h′,k′)h′=1−3/4m⇒34m=1−h′⇒m=34(1−h′)k′=34−m⇒m=(34−k′)m=34(1−h′)=34−k′⇒31−h′=3−4k′⇒3=3(1−h′)−4k′(1−h′)⇒3h′+4k′=4h′k′⇒3x+4y=4xy

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