The locus of the midpoints of the chords of contact of `x^2+y^2=2` from the points on the line `3x+4y=10` is a circle with center `Pdot` If `O` is the origin, then `O P` is equal to 2 (b) 3 (c) `1/2` (d) `1/3`

Options

A. 2
B. 3
C. `1//2`
D. `1//3`

Answer

3

Explanation

Chord with midpoint (h,k) is
`hx+ky=h^(2)+k^(2)` (1)
Chord of contact of `(x_(1),y_(1))` is
`x x_(1)+y y_(1)=2` (2)
Comparing (1) and (2), we get
`x_(1)=(2h)/(h^(2)+k^(2))` and `y_(1)=(2k)/(h^(2)+k^(2))`
Since `(x_(1),y_(1))` lies on `3x+4y=10,6h+8k=10(h^(2)+k^(2))`.
Therefore, the locus of (h,k) is
`x^(2)+y^(2)-(3)/(5)x-(4)/(5)y=0`
which is a circle with center `P(3//10,4//10)`.Therefore, `OP = 1//2`