a variable line `L` is drawn trough `O(0,0)` to meet the lines `L_1:y-x-10=0` and `L_2:y-x-20=0` at point `A&B` respectively .A point `P` is taken on line `L` the `(1) ` if `2/(OP)=1/(OA)+1/(OB)` then locus of `P` is `(2)` if `(OP)^2=(OA)*(OB)` then locus of `P` is `(3)` if `1/(OP)^2=1/(OA)^2+1/(OB)^2` then locus of point `P` is:

Options

A. `(y-x)^(2) = 80`
B. `(y-x)^(2) = 100`
C. `(y-x)^(2) = 64`
D. none of these

Answer

A

Explanation

`(1)/(r^(2)) = ("sin" theta - "cos" theta)^(2)/(100) + ("sin" theta -"cos" theta)^(2)/(400)`
`"or " 400 = 5(r "sin" theta - r "cos" theta)^(2)`
Hence, the locus is `400=5(x-y)^(2), i.e., (x-y)^(2)= 80.`