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Question

The tangent at a point P on the curve xmyn=am+n meets the x-axis at A and y-axis at B. The triangle formed by AOB has centroid at Q, If O be the origin, then

A
AP:PB=n:m
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B
AP:PB=m:n
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C
locus of Q is (3xm)m(3yn)n=[a(m+n)]m+n
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D
locus of Q is (3xm)n(3yn)m=[a(m+n)]m+n
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Solution

The correct options are
A AP:PB=n:m
C locus of Q is (3xm)m(3yn)n=[a(m+n)]m+n
xmyn=am+nmlnx+nlny=(m+n)lna
Differentiating w.r.t. x
mx+nyy=0y=mynx
So, equation of the tangent at P(x,y) is
Yy=mynx(Xx)
Tangent meets the x-axis at A and y-axis at B.
A=((m+n)xm,0)B=(0,(m+n)yn)

Let P divides AB internally in the ratio of r:1
Then x=r0+1(m+n)xmr+1
(r+1)x=(1+nm)xr=nmAP:PB=n:m

Let coordinates of Q is (h,k)
h=(m+n)x3m
and k=(m+n)y3n
(3hm)m(3kn)n=(m+n)m+nxmyn(3xm)m(3yn)n=[a(m+n)]m+n

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