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Circle question

2004 · Shift 0 · Q110
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Circle question

2004 · Shift 0 · Q110

JEE MainMathematicsCircleMCQ+4 / −1
If a circle passes through the point (a, b) and cuts the circle x2 + y2=4{x^2}\, + \,{y^2} = 4x2+y2=4 orthogonally, then the locus of its centre is :
  1. A
    2ax −2by −(a2 + b2+4)=02ax\, - 2by\, - ({a^2}\, + \,{b^2} + 4) = 02ax−2by−(a2+b2+4)=0
  2. B
    2ax +2by −(a2 + b2+4)=02ax\, + 2by\, - ({a^2}\, + \,{b^2} + 4) = 02ax+2by−(a2+b2+4)=0
  3. C
    2ax −2by +(a2 + b2+4)=02ax\, - 2by\, + ({a^2}\, + \,{b^2} + 4) = 02ax−2by+(a2+b2+4)=0
  4. D
    2ax +2by +(a2 + b2+4)=02ax\, + 2by\, + ({a^2}\, + \,{b^2} + 4) = 02ax+2by+(a2+b2+4)=0
View written solutionFree

Correct answer: B

  1. Let the required circle have centre (h,k)(h,k)(h,k) and radius rrr.

    Its equation is (x−h)2+(y−k)2=r2.(x-h)^2+(y-k)^2=r^2.(x−h)2+(y−k)2=r2.

  2. Since the circle passes through (a,b)(a,b)(a,b), we have (a−h)2+(b−k)2=r2.(a-h)^2+(b-k)^2=r^2.(a−h)2+(b−k)2=r2.

  3. Condition for orthogonal intersection of two circles

    The given circle is x2+y2=4,x^2+y^2=4,x2+y2=4, whose centre is (0,0)(0,0)(0,0) and radius is 222.

    If two circles with centres distance ddd and radii r1,r2r_1,r_2r1​,r2​ cut orthogonally, then d2=r12+r22.d^2=r_1^2+r_2^2.d2=r12​+r22​.

    Here,

    \, r_1=r, \, r_2=2. $$ So, $$ h^2+k^2=r^2+4. $$

  4. Substitute r2=(a−h)2+(b−k)2r^2=(a-h)^2+(b-k)^2r2=(a−h)2+(b−k)2:

    h2+k2=(a−h)2+(b−k)2+4.h^2+k^2=(a-h)^2+(b-k)^2+4.h2+k2=(a−h)2+(b−k)2+4.

  5. Expand the right-hand side:

    h2+k2=a2+h2−2ah+b2+k2−2bk+4.h^2+k^2=a^2+h^2-2ah+b^2+k^2-2bk+4.h2+k2=a2+h2−2ah+b2+k2−2bk+4.

    Cancelling h2+k2h^2+k^2h2+k2 from both sides, 0=a2+b2+4−2ah−2bk.0=a^2+b^2+4-2ah-2bk.0=a2+b2+4−2ah−2bk.

    Rearranging, 2ah+2bk−(a2+b2+4)=0.2ah+2bk-(a^2+b^2+4)=0.2ah+2bk−(a2+b2+4)=0.

  6. Hence the locus of the centre (h,k)(h,k)(h,k) is 2ax+2by−(a2+b2+4)=0.2ax+2by-(a^2+b^2+4)=0.2ax+2by−(a2+b2+4)=0.

  7. Matching with options

    This is Option B.

Verification with stored answer: Stored correct answer is B, which matches the derived answer.

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