CircleHardBloom L3
Question
A circle with center $(2,2)$ touches the coordinate axes and a straight line $AB$ where $A$ and $B$ lie on positive direction of coordinate axes such that the circle lies between origin and the line $AB$. If $O$ be the origin then the locus of circumcenter of $\bigtriangleup OAB$ will be:
Options
A.$xy = x + y + \sqrt{x^{2} + y^{2}}$
B.$xy = x + y - \sqrt{x^{2} + y^{2}}$
C.$xy + x + y = \sqrt{x^{2} + y^{2}}$
D.$xy + x + y + \sqrt{x^{2} + y^{2}} = 0$
Solution
[Key Concept: A circle touching both axes has centre $(r,r)$; the cut-off tangent triangle uses the tangent length from the corner.]
$r = \frac{\Delta}{s} = \frac{\frac{1}{2} \times (2h)(2k)}{\frac{1}{2}\left( 2h + 2k + 2\sqrt{h^{2} + k^{2}} \right)}$
$2 = \frac{2hk}{h + k + \sqrt{h^{2} + k^{2}}}$
$
$⇒ Locus of $xy = x + y + \sqrt{x^{2} + y^{2}}$
Create a free account to view solution
View Solution FreeMore Circle Questions
Number of points (x, y) having integral coordinates satisfying the condition x2 + y2 < 25 is -...If (− 3, 2) lies on the circle x2 + y2 + 2gx + 2fy + c = 0 which is concentric with the circle x2 + y2 + 6x + 8y &...The locus of the centers of the circles which cut the circles x2 + y2 + 4x - 6y + 9 = 0 and x2 + y2 - 5x + 4y - 2 = 0 or...The gradient of the tangent line at the point (a cos α, asin α ) to the circle x2 + y2 = a2, is -...The length of the shortest path that be gives at the point (2,5), touches the x-axis and then ends at a point on the cir...