Question

For real values the expression $x^2-xy+y^2-4x-4y+16=0$ represents

• A. A point.
• B. A circle.
• C. A pair of straight lines.
• D. None of the above.

Answer 1

The correct option is A A point.

$x^2-xy+y^2-4x-4y+16=0⟹x^2-(y+4)x+y^2-4y+16=0$

since $x$ is real, so $\Delta \ge 0$.

or ${(y+4)}^2-4(y^2-4y+16)\ge 0⟹-3y^2-24y+48\ge 0⟹y^2-8y+16\le 0⟹{(y-4)}^2\le 0⟹y-4=0⟹y=4$

since equation is symmmetric ,

$\therefore$ $x=4$

so a point.