Let `2 x^2 + y^2 - 3xy = 0` be the equation of pair of tangents drawn from the origin to a circle of radius 3, with center in the first quadrant. If A is the point of contact. Find OA

Answer

`3 ( 3+sqrt(10))`

Explanation

The equation o`2x^(2)-3xy+y^(2)=0` represents a pair of tangents OA and `OA'`.
Let the angle between these two tangents be `2 theta`. Then,
` tan 2 theta =(2 sqrt((-3//2)^(2)-2xx1))/(2+1)`

or `(2 tan theta)/(1-tan^(2)theta)=(1)/(3)`
or `tan^(2) theta +6 tan theta -1=0`
or `tan theta = (-6 +- sqrt(36+4))/(2)= -3 +- sqrt(10)`
Now, we know that the line joining the point through which tangents are drawn to the center bisects the angle between the tangents. Therefore,
`/_AOC = theta`
In `Delta OAC`,
`tan theta =(3)/(OA)`
or `OA =(3)/(sqrt(10)-3) xx (sqrt(10)+3)/(sqrt(10)+3)`
`:. OA =3(3+sqrt(10))`