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A point P moves on x y - plane such that P S + P S ' = 4 , where S ( K ,   0 ) and S ' ( - K ,   0 ) , then which of the following is not true about the locus of P   ?

Options

  1. AEllipse if K ∈ ( - 2 ,   2 )
  2. BLine segment if K = ± 2
  3. CEmpty if K ∈ ( - ∞ ,   - 2 ) ∪ ( 2 ,   ∞ )
  4. DNone of these

Correct answer

D. None of these

Step-by-step solution

Given, S ( K ,   0 ) and S ' ( - K ,   0 ) , Now, S S ' = | 2 K | and P S + P S ' = 4 (given) Now for the ellipse we know that, P S + P S ' > S S ' ⇒ 4 > | 2 K | ⇒ K ∈ ( - 2 ,   2 ) . If P , S   &   S ' are collinear, then P S + P S ' = S S ' ⇒ K = ± 2 . If P S + P S ' < S S ' , then no locus exists. ⇒ K ∈ ( - ∞ ,   - 2 ) ∪ ( 2 ,   ∞ ) .

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