Question

The equation of circle A is x2 + y2 + Cx + Dy + E = 0. If the circle is moved horizontally to the left of the y-axis without changing the radius, how are the coefficients C and D affected?

Answer 1

Option E.

Explanation:

E. C and D are unchanged, but E increases.

Answer 2

You can complete the square to find the equation of the circle in standard form.


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


`\frac{C^2+D^2}4=x^2+Cx+\frac{C^2}4+y^2+Dy+\frac{D^2}4+E`


`\frac{C^2+D^2}4-E=\left(x+\frac C2\right)^2+\left(y+\frac D2\right)^2`

So the circle is centered at
`\left(-\frac C2,-\frac D2\right)` and has radius
`\sqrt{\frac{C^2+D^2}4-E}`.

A horizontal shift to the left can be done by adding a positive number to the
`x` term, as in


`\frac{C^2+D^2}4-E=\left(x+\frac C2+\mathbf k\right)^2+\left(y+\frac D2\right)^2`

Expanding, you end up with


`x^2+(C+2k)x+y^2+Dy+E+Ck=0`

So the coefficient
`C` is increased by twice the horizontal shift, while
`D` remains unchanged.