Question

The equation of the circle which touches both axes and whose centre is (x1, y1 ) is

Answer 1

Answer:   The equation of the circle is $x^2+y^2-2x_1(x+y)+x_1^2=0.$

Step-by-step explanation: The equation of a circle in standard form is given by

$(x-x_1)^{2}+(y-y_1)^{2}=r^2,$ where $(x_1,y_1)$ is the centre of the circle and $r$ is the radius.

In the attached figure, a circle with centre O(1,1) is shown touching both the axes. Here, the circle touches the X-axis at A(1,0) and Y-axis at B(0,1). Also, the radius is 1 unit.

Since the centre is given to be $(x_1,y_1)$, so the circle touches the X-axis at $(x_1,0)=(y_1,0)$ and Y-axis at $(0,x_1)=(0,y_1).$

In short, $x_1=y_1= \textup{radius of the circle}.$

So, the equation of the circle is

$(x-x_1)^2+(y-x_1)^2=x_1^2\\\\\Rightarrow x^2+y^2-2x_1(x+y)+x_1^2=0.$