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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Let the set of all values of $r$, for which the circles
$(x + 1)^2 + (y + 4)^2 = r^2$ and $x^2 + y^2 - 4x - 2y - 4 = 0$
intersect at two distinct points be the interval $(\alpha, \beta)$. Then $\alpha\beta$ is equal to
Solution
$(x-2)^2 + (y-1)^2 = 9$ & $(x+1)^2 + (y+4)^2 = r^2$
$r_1 < c < r_1 + r_2$
$|r-3| < \sqrt{(2+1)^2 + (1+4)^2} < r + 3$
$|r-3| < \sqrt{34} < r + 3$
$-\sqrt{34} < r-3 < \sqrt{34}$
$\Rightarrow r \in (3-\sqrt{34}, 3+\sqrt{34})$
$\therefore \alpha\beta = (3-\sqrt{34})(3+\sqrt{34}) = 9 - 34 = -25$
$\Rightarrow 25$
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