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AP EAMCET202423 May 2024Morning ShiftMathematicsStraight LinesActual

The locus of the midpoint of the portion of the line x +y =p intercepted by the coordinate axes, where p is a constant, is

Options

  1. A1 x^2 + 1 y^2 = 3 p^2
  2. B1 x^2 + 1 y^2 = 4 p^2
  3. Cx^2+y^2=2 p^2
  4. D2 x^2 + 2 y^2 = 1 p^2

Correct answer

B. 1 x^2 + 1 y^2 = 4 p^2

Step-by-step solution

Given, x +y =p ...(i) Let P(h, k) be the mid point of above line. When x=0 , Eq. (i) becomes y =p y= p When y=0 , Eq. (i) becomes x =p x= p The line meets the coordinate axes at A ( p , 0 ), B (0, p ) Midpoint P(h, k)= ( p 2 , p 2 ) = p 2 h and = p 2 k Squaring and adding the above equations, ^2 + ^2 = p^2 4 h^2 + p^2 4 k^2 4 p^2 = 1 h^2 + 1 k^2 Locus of P is 1 x^2 + 1 y^2 = 4 p^2 .

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