If x - y = 2 is the equation of a chord of the circle x ^ 2 + y ^ 2 + 2y = 0 find the equation of the circle of which this chord is a diameter. Which circle occupies more area and why?

 If x - y = 2 is the equation of a chord of the circle x ^ 2 + y ^ 2 + 2y = 0 find the equation of the circle of which this chord is a diameter. Which circle occupies more area and why?

Solutions

Given that

• the equation of the chord $x-y=2.....\left[1\right]$

• the equation of the circle $x^2+y^2+2y=0......\left[2\right]$

From these two equations we get

${(y+2)}^2+y^2+2y=0$

$\Rightarrow 2y^2+6y+4=0$

$\Rightarrow 2y^2+6y+4=0$

$\Rightarrow y^2+3y+2=0$

$\Rightarrow y^2+2y+y+2=0$

$\Rightarrow y(y+2)+1(y+2)=0$

$\Rightarrow (y+2)(y+1)=0$

So $y=-1\text{and}-2$

Inserting in [1] we get

$x=1\text{and}0$ respectively

So the coordinates of points intersections of the chord with the circles are $(1,-1)\text{and}(0,-2)$
The line segment of the chord is also the diameter of a circle.

So equation of this circle having diameter $x-y=2$ will be

$\frac{y+1}{x-1}\times \frac{y+2}{x-0}=-1$

$\Rightarrow (y+1)\times (y+2)=-x(x-1)$

$\Rightarrow x^2+y^2-x+3y+2=0.....\left[3\right]$

This is the required equation of the circle