AP EAMCET · Maths · Circle

Let $C$ be the circle with centre $(0, 0)$ and radius $3 unit$ . The equation of the locus of the mid points of the chords of the circle $C$ that subtend an angle of $\frac{2 π}{3}$ at its centre, is:

A $x^{2} + y^{2} = 1$
B $x^{2} + y^{2} = \frac{27}{4}$
C $x^{2} + y^{2} = \frac{9}{4}$
D $x^{2} + y^{2} = \frac{3}{2}$

Verified Solution

Answer & Solution

Correct Answer

(C) $x^{2} + y^{2} = \frac{9}{4}$

Step-by-step Solution

Detailed explanation

Let the coordinates of a point P be (h, k) which is mid point of the chord AB. Now, OP=h-02+k-02 =h2+k2 In ∆AOP, cos⁡π3=OPOA ⇒ 12=h2+k23 ⇒ h2+k2=94 Hence, the required locus is x2+y2=94.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let C be the circle with centre 0, 0 and radius 3 unit. The equation of the locus of the mid points of the chords of the circle C that subtend an angle of 2π3 at its centre, is:

Step-by-step Solution

Let $C$ be the circle with centre $(0, 0)$ and radius $3 unit$ . The equation of the locus of the mid points of the chords of the circle $C$ that subtend an angle of $\frac{2 π}{3}$ at its centre, is: