AP EAMCET · Maths · Straight Lines

Let $A, B$ and $C$ be three points in a plane. The locus of a point $P$ moving such that $P A^{2} + P B^{2} = 2 P C^{2}$ is a

A straight line
B pair of straight lines
C circle
D parabola

Verified Solution

Answer & Solution

Correct Answer

(A) straight line

Step-by-step Solution

Detailed explanation

We have A, B and C be three points in a plane. Let A≡0,1, B≡1,0 & C≡-1,0 Let Px,y be any variable point. Then we have PA2+PB2=2PC2 ⇒x-02+y-122+x-12+y-022=2x+12+y-022 ⇒x2+y-12+x-12+y2=2x+12+y2 ⇒6x+2y=0 It is a linear…

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 A, B and C be three points in a plane. The locus of a point P moving such that PA2+PB2=2PC2 is a

Step-by-step Solution

Let $A, B$ and $C$ be three points in a plane. The locus of a point $P$ moving such that $P A^{2} + P B^{2} = 2 P C^{2}$ is a