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BITSAT 2012
Exam
BITSAT
Year
2012
Subject
Mathematics
Chapter
Circle

BITSAT 2012 · Mathematics · Circle

If the line x α+y α=p represents the common chord of the circles x^...

If the line xcos⁡α+ysin⁡α=px \cos \alpha+y \sin \alpha=p represents the common chord of the circles x2+y2=a2x^{2}+y^{2}=a^{2} and \mathrmx2+\mathrmy2+\mathrmb2(a>b),\mathrmx^{2}+\mathrmy^{2}+\mathrmb^{2}(\mathrm{a} > \mathrm{b}), where A\mathrm{A} and B\mathrm{B} lie ont he first circle and P\mathrm{P} and Q\mathrm{Q} lie on the second circle, then AP is equal toIf the line x α+y α=p represents the common chord of the circles x^ 2 +y^ 2 =a^ 2 and ^ 2 + ^ 2 + ^ 2 (a > b), where A

Options

  1. Aa2+p2+b2+p2\sqrt{a^{2}+p^{2}}+\sqrt{b^{2}+p^{2}}
  2. Ba2−p2+b2−p2\sqrt{a^{2}-p^{2}}+\sqrt{b^{2}-p^{2}}
  3. Ca2−p2−b2−p2\sqrt{a^{2}-p^{2}}-\sqrt{b^{2}-p^{2}}
  4. Da2+p2−b2+p2\sqrt{a^{2}+p^{2}}-\sqrt{b^{2}+p^{2}}

Correct answer

C. a2−p2−b2−p2\sqrt{a^{2}-p^{2}}-\sqrt{b^{2}-p^{2}}

Step-by-step solution

The given circles are concentric with centre at (0,0) and the length of the perpendicular from (0,0) on the given line is p\mathrm{p}. Let OL=p\mathrm{OL}=\mathrm{p} then, AL=OA2−OL2=\mathrma2−\mathrmp2\mathrm{AL}=\sqrt{\mathrm{OA}^{2}-\mathrm{OL}^{2}}=\sqrt{\mathrma^{2}-\mathrmp^{2}} and PL=OP2−OL2=\mathrmb2−\mathrmp2⇒AP=\mathrma2−\mathrmp2−\mathrmb2−\mathrmp2\mathrm{PL}=\sqrt{\mathrm{OP}^{2}-\mathrm{OL}^{2}}=\sqrt{\mathrmb^{2}-\mathrmp^{2}}\Rightarrow \mathrm{AP}=\sqrt{\mathrma^{2}-\mathrmp^{2}}-\sqrt{\mathrmb^{2}-\mathrmp^{2}} If the line x α+y α=p represents the common chord of the circles x^ 2 +y^ 2 =a^ 2 and ^ 2 + ^ 2 + ^ 2 (a > b), where A
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