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Question

The locus of the foot of the perpendicular drawn from origin to a straight line which passes through a fixed point P(h,k) is denoted by S. The tangent at P(h,k) on the curve S is given by

A
xh+ykk2=0
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B
xh+ykh2=0
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C
2xh+2ykh2k2=0
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D
xh+ykh2k2=0
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Solution

The correct option is D xh+ykh2k2=0
The equation of a variable line passing through P(h,k) is given by yk=m(xh)
If (α,β) is the foot of the perpendicular drawn from the origin to the line, then
β0α0×m=1
m=αβ
βk=m(αh)
βk=αβ(αh)
α2+β2αhβk=0

x2+y2xhyk=0 is locus which is a circle.
By point form, tangent equation is
xh+ykh(x+h2)k(y+k2)=0
xh+ykh2k2=0 is the equation of tangent

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