1.

A point P moves such that the sum of the slopes of the normals drawn from it to the hyperbola xy = 16 is equal to the sum of ordinates of feet of normals . The locus of P is a curve C. If the tangent to the curve C cuts the corrdinate axes at A and B, then the locus of the middle point of AB is

Answer»

`x^(2)=4y`
`x^(2)=2y`
`x^(2)+2y=0`
`x^(2)+4y=0`

Solution :`x^(2)=16Y`
The equation of TANGENT of P is

`x*4t=(16(y+t^(2)))/(2)`
`"or"4txx=8y+8t^(2)`
`"or"tx=2y+2t^(2)`
`A-=(2t,0), B-=(0,t^(2))` LTBRGT M(H,k) is the middle point of AB.
`h=t,k=-(t^(2))/(2)or2k-h^(2)`
Therefore, the locus of `M(h,k)` is `x^(2)+2y=0`.


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