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Let S be the circle in the x y -plane de...

Let `S` be the circle in the `x y` -plane defined by the equation `x^2+y^2=4.` (For Ques. No 15 and 16) Let `P` be a point on the circle `S` with both coordinates being positive. Let the tangent to `S` at `P` intersect the coordinate axes at the points `M` and `N` . Then, the mid-point of the line segment `M N` must lie on the curve (a)`(x+y)^2=3x y` (b) `x^(2//3)+y^(2//3)=2^(4//3)` (c) `x^2+y^2=2x y` (d) `x^2+y^2=x^2y^2`

A

`(x+y)^(2) = 3xy`

B

`x^(2//3)+y^(2//3) =2^(4//3)`

C

`x^(2)+y^(2)=2xy`

D

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

Text Solution

Verified by Experts

The correct Answer is:
4
Let the coordinates of P be `(2 cos theta , 2 sin theta )`
Equation of tangent to circle at P is `x cos x + y sin theta =2`
`:. M((2)/(cos theta ,0)),N(0,(2)/(cos theta))`
Let the mid -point of MN be `(x,y)`
`:. x= (1)/(cos theta ) ` and `y = (1)/(sin theta)`
Squaring and adding , we get
`(1)/(x^(2))+(1)/(y^(2))=1 `
or `x^(2)+y^(2)=x^(2)y^(2)`, this is the required locus
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