Multiple choice

STATEMENT - 1 : Locus of mid point of chords of circle $x^{2} + y^{2} = 4$ which subtends angle of $\displaystyle \frac{\pi }{2}$ at origin is $x^{2} + y^{2} = 1.$ STATEMENT - 2 : If any chord of circle $x^{2} + y^{2} = r^{2}$ subtends an angle $'\theta '$ at center, then its mid point always lies on $x^{2} + y^{2} = r^{2} \cos^{2} \left ( \displaystyle \frac{\theta }{2} \right )$

  1. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

  2. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

  3. STATEMENT-1 is True, STATEMENT-2 is False

  4. STATEMENT-1 is False, STATEMENT-2 is True
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D Correct answer
Explanation

Statement 2 is a known theorem: the locus of the midpoint of a chord subtending angle theta at the center is a circle of radius r*cos(theta/2). For theta=pi/2, radius = r*cos(pi/4) = r/sqrt(2). Here r=2, so radius = 2/sqrt(2) = sqrt(2). The locus is x^2 + y^2 = 2. Statement 1 claims x^2 + y^2 = 1, which is false.

AI explanation

For a circle given by x^2 + y^2 = r^2, the distance d from the center to a chord that subtends an angle theta can be found using the right triangle formed by the radius, half the chord, and the perpendicular distance. This yields the relation d = r * cos(theta / 2). For the specific circle x^2 + y^2 = 4 (where r = 2) and an angle of pi/2, the distance is 2 * cos(pi/4), which results in sqrt(2). Therefore, the correct locus of the midpoint is the circle x^2 + y^2 = 2, making Statement 1 false while Statement 2 remains a true general property.

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