First slide

Definition of a circle

Question

The equation ${(x^{2} - a^{2})}^{2} + {(y^{2} - b^{2})}^{2} = 0$ represents points

Moderate

A

which are collinear
B

which lie on a circle with centre at $(0, 0)$
C

which lie on a circle with centre at $(a, b)$
D

none of these

Solution

We have
$\begin{array}{l} {(x^{2} - a^{2})}^{2} + {(y^{2} - b^{2})}^{2} = 0 \\ \Rightarrow x^{2} - a^{2} = 0 and y^{2} - b^{2} = 0 \Rightarrow x = \pm a and y = \pm b . \end{array}$
Thus, the points are $(a, b), (a, - b), (- a, b)$ and $(- a, - b) .$
Clearly, these points lie on a circle $x^{2} + y^{2} = a^{2} + b^{2}$ having centre at the origin.

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