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Question

$$f(x, y) = x^2 + y^2 + 2ax + 2by + c = 0$$ represents a circle. If $$f(x, 0) = 0$$ has equal roots, each being $$2$$ and $$f(0, y) = 0$$ has $$2$$ and $$3$$ as it's roots, then centre of circle is


A
(2,52)
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B
Data are not sufficient
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C
(2,52)
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D
Data are inconsistent
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Solution

The correct option is D Data are inconsistent
$$x^2 +2ax + c = (x-2)^2$$
$$\Rightarrow$$                 $$-2a = 4, c = 4$$
$$\Rightarrow$$                 $$a = -2, c = 4$$
                 $$y^2 + 2by + c = (y-2)(y-3)$$
$$\Rightarrow$$                 $$-2b = 5, c = 6$$
$$\Rightarrow$$      $$b = -\dfrac{5}{2}, c = 6$$ clearly the data are not consistent.

Mathematics

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