Solve x^2+y^2-8x+10y+k=0

$$x ^ { 2 } + y ^ { 2 } - 8 x + 10 y + k = 0$$

Solve for k

$$k=-x^{2}+8x-y^{2}-10y$$

Solution Steps

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$y^{2}-8x+10y+k=-x^{2}$$

Subtract y^{2} from both sides.

$$-8x+10y+k=-x^{2}-y^{2}$$

Add 8x to both sides.

$$10y+k=-x^{2}-y^{2}+8x$$

Subtract 10y from both sides.

$$k=-x^{2}-y^{2}+8x-10y$$

Solve for x (complex solution)

$$x=\sqrt{16-k-10y-y^{2}}+4$$

$$x=-\sqrt{16-k-10y-y^{2}}+4$$

Solve for x

$$x=\sqrt{16-k-10y-y^{2}}+4$$

$$x=-\sqrt{16-k-10y-y^{2}}+4\text{, }k\leq 16-10y-y^{2}$$