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10.1 circle and system of circle — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS10.1 circle and system of circle1 Mark
MCQ
If the circles ${x^2} + {y^2} = 4,{x^2} + {y^2} - 10x + \lambda = 0$ touch externally, then $\lambda $ is equal to 
  • ✓
    $-16$
  • B
    $9$
  • C
    $16$
  • D
    $25$
✓

Answer

Correct option: A.
$-16$
a
(a) Circles $x^2$ + $y^2$ = 4, $x^2$ + $y^2$ -10x + $\lambda$ = $0$ touch externally

${C_1}{C_2} = {r_1} + {r_2}$

==> ${C_1}(0,\,0)$ and ${C_2} = (5,\,0)$

${r_1} = 2$ and ${r_2} = \sqrt {25 + \lambda } $

 $\sqrt {{{(5 - 0)}^2} + 0} = 2 + \sqrt {25 + \lambda } $

==> $5 - 2 = \sqrt {25 + \lambda } $

==> $3 = \sqrt {25 + \lambda } $

==> $9 = 25 + \lambda $

==> $\lambda$ $= -16.$

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