Question

The equation of a circle is x2 + y2 + Cx + Dy + E = 0. If the radius of the circle is decreased without changing the coordinates of the center point, how are the coefficients C, D, and E affected?

Answer 1

Answer: C and D will be unaffected while E will increase.

Explanation:

Given equation of a circle is,

`x^2+y^2+Cx+Dy+E=0`

But we know that the equation of the circle,

`(x-h)^2+(y-k)^2=r^2`

Where, (h,k) is the center of circle and r is the radius of circle,

We know that, (a-b)² = a² - 2ab + b²,

`\implies x^2-2hx+h^2+y^2-2yk+k^2=r^2`

`\implies x^2+y^2-2hx-2yk+(h^2+k^2-r^2)=0`

By comparing,

`C=-2h`,

`D=-2y`,

`E=h^2+k^2-r^2`

Thus, by the above values, we can say that,

If r decreases C will be unaffected ( because C is free from r)

D will be unaffected ( because D is also free from r)

While, E will be increased.

Hence, C and D will be unaffected while E will increase if radius decreases.

Answer 2

There will be no change in C and D but E must increase.

Explanation:

We are given a equation of circle as:

`x^2+y^2+Cx+Dy+E = 0`

it could also be represented in general form of equation of circle as:

`(x+(C)/(2))^2+(y+(D)/(2))^2=(C^2)/(4)+(D^2)/(4)-E`

( since we add both the side of the equality by
`(C^2)/(4)` and
`(D^2)/(4)` )

Hence, on comparing the equation with general form of the equation:

`(x-g)^2+(y-h)^2=r^2`

The center of circle is (g,h) and radius is r.

Here we have the center as:

`((-C)/(2),(-D)/(2))`

and radius is given aS:

`r=\sqrt{(C^2)/(4)+(D^2)/(4)-E}\\\\r=\sqrt{(C^2+D^2-4E)/(4)`

`r=(√(C^2+D^2-4E))/(2)`

As there is no change in the center hence the value of
`(-C)/(2)` and
`(-D)/(2)` remains unchanged i.e. the value of C and D remains unchanged.

Now we are given that the radius of the circle is decreased.

that means the change in radius is due to the change in E.

Hence for the quantity
`r=(√(C^2+D^2-4E))/(2)\\` to decrease E must increase so that the total quantity decreases.

Hence there will be no change in C and D but E must increase in order to decrease the radius of circle.