Multiple choice

If the equation of the locus of the mid-point of the chord which subtends right angle at the centre of a circle is x2 + y2 = 9, then find the equation of the circle on which this chord exists.

  1. x2 + y2 = 36

  2. 2x2 + 2y2 = 9

  3. x2 + y2 = 18

  4. (x - 1)2 + (y - 2)2 = 9

  5. None of these
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C Correct answer
Explanation

For a chord of a circle to subtend a right angle at the center, the distance from the center to the chord must be r/sqrt(2). If the locus of the midpoint is x^2 + y^2 = 9, the radius of this locus circle is 3. Since the distance from the center to the chord is 3, the radius of the original circle is 3 * sqrt(2). Thus, the equation is x^2 + y^2 = (3 * sqrt(2))^2 = 18.

AI explanation

If a chord subtends a right angle at the center of a circle with radius R, the distance d from the center to the midpoint of the chord is R times the cosine of 45 degrees, making d equal to R divided by the square root of 2. The locus of this midpoint is given as a circle with radius d equal to 3. Setting R divided by the square root of 2 equal to 3 gives R equal to 3 times the square root of 2, and squaring this radius gives the equation of the original circle. The equation of the original circle is x squared plus y squared equals 18.

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