The locus of foot of perpendicular from the center of the hyperbola xy=c^2 on a variable tangent is (1){x^2-y^2}=4c^2xy (2){x^2+y^2}=2c^2xy (3){x^2+y^2}=4c^2xy (4){x^2+y^2}=4c^2xy{plese explen}

Hi student,
we have rectangular hyperbola
xy=c²

take a general point on this in parametric P(ct, c/t)
Now tangent at point P is T=0
xy₁ + yx₁ = 2c²
So in parametric
x(c/t) + y(ct) = 2c²
x/t + ty – 2c = 0….….…...(1)

Now centre is O(0,0)
LET foot of Perpendiculer be F(h,k)

So OF tangent
So product of slopes = -1
k/h * (-1/t²)= -1
hence k/h= t² ….…..(2)

Also prependicular distance OF is

Now Put t²=k/h in above you will get the answer

I hope the solution is clear