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10-1.Circle and System of Circles
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यदि वृत्त ${x^2} + {y^2} + 2ax + c = 0$ तथा ${x^2} + {y^2} + 2by + c = 0$ एक-दूसरे को स्पर्श करते हों तो
A
$\frac{1}{a} + \frac{1}{b} = \frac{1}{c}$
B
$\frac{1}{{{a^2}}} + \frac{1}{{{b^2}}} = \frac{1}{{{c^2}}}$
C
$\frac{1}{a} + \frac{1}{b} = {c^2}$
D
$\frac{1}{{{a^2}}} + \frac{1}{{{b^2}}} = \frac{1}{c}$
Solution
(d) ${C_1}( – a,\;0);$${C_2}(0,\; – b);$
${R_1}(\sqrt {{a^2} – c} );$ ${R_2}(\sqrt {{b^2} – c} )$
${C_1}{C_2} = \sqrt {{a^2} + {b^2}} $
चूँकि ये एक-दूसरे को स्पर्श करते हैं, अत:
$\sqrt {{a^2} – c} + \sqrt {{b^2} – c} = \sqrt {{a^2} + {b^2}} $
$ \Rightarrow {a^2}{b^2} – {b^2}c – {a^2}c$= 0
$\frac{1}{{{a^2}{b^2}c}}\;$ से गुणा करने पर,
$\frac{1}{{{a^2}}} + \frac{1}{{{b^2}}} = \frac{1}{c}$.
Standard 11
Mathematics
