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Question

# A ladder 12 units long slides in a vertical plane with its ends in contact with a vertical wall and a horizontal floor along x−axis. Then the locus of a point on the ladder 4 units from its foot, is

A
x264+y232=1
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B
x264+y216=1
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C
x232+y232=1
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D
x28+y24=1
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Solution

## The correct option is B x264+y216=1Given Let QR is a ladder, in which Q≡(a,0) and R≡(0,b) Given, a2+b2=122=144…(i) Clearly P divides QR internally in the ratio 1:2 ∴ By section formula, P is of the form. P=(x,y)=(mx2+nx1m+n,my2+ny1m+n) Here, m=1,n=2 Q=(a,0)=(x1,y1) and R=(0,b)=(x2,y2) ∴P=(x,y)=(1×0+2×a1+2,1×b+2×01+2) i.e., (x,y)=(2a3,b3) ∴x=2a3,y=b3 ⇒h=2a3 and k=b3 ⇒a=3h2 and b=3k From (i), we get 9h24+9k2=144 ⇒x264+y216=1

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