For how many values of , the circle and the coordinate axes have exactly three common points? [2017]

Question

For how many values of $p$ , the circle $x^{2} + y^{2} + 2 x + 4 y - p = 0$ and the coordinate axes have exactly three common points? [2017]

Smart Achievers · Accepted Answer

(2)

$Centre of the circle is (- 1, - 2)$

Geometrically, circle will have exactly 3 common points with axes in the cases

$(i) Passing through origin \Rightarrow p = 0$

$(ii) Touching x -axis and intersecting y -axis at two points$ $i.e. f^{2} > c and g^{2} = c$

$i.e. 4 > - p and 1 = - p \Rightarrow p > - 4 and p = - 1$ $∴$ $p = - 1$

$(iii) Touching y -axis and intersecting x -axis at two points$ $i.e. f^{2} = c and g^{2} > c$

$\Rightarrow 4 = - p and 1 > - p$

$\Rightarrow p = - 4 and p > - 1, which is not possible$

$∴$ $Only two values of p are possible.$

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