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The locus of the point of intersection of the lines 2 x - y + 4 2 k = 0 and 2 k x + k y - 4 2 = 0 ( k is any non-zero real parameter) is

Options

  1. Aan ellipse whose eccentricity is 1 3
  2. Ba hyperbola whose eccentricity is 3
  3. Ca hyperbola with length of its transverse axis 8 2
  4. Dan ellipse with length of its major axis 8 2

Correct answer

C. a hyperbola with length of its transverse axis 8 2

Step-by-step solution

Given equation of lines are 2 x - y + 4 2 k = 0           . . . 1 and 2 k x + k y - 4 2 = 0             . . . 2 Eliminating k from 1 and 2 , ⇒ 2 x + y 2 x - y - 4 2 = 4 2 ⇒ 2 x 2 - y 2 = - 32 ⇒ y 2 32 - x 2 16 = 1 , which is a hyperbola. ⇒ e = 1 + 16 32 = 3 2 and length of transverse axis = 8 2 .

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