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NEST2026MathematicsStraight Lines

Let a variable line L meet x -axis and y -axis at points A and B, respectively. Suppose the distance of the line L from the origin is 3 units. Then the equation of the locus of the point C that divides the line segment AB internally in the ratio 2 : 1 is

Options

  1. Ax^2 + 4y^2 = 9
  2. B4 x^2 + 1 y^2 = 1
  3. C1 x^2 + 4 y^2 = 1
  4. D4x^2 + y^2 = 9

Correct answer

C. 1 x^2 + 4 y^2 = 1

Step-by-step solution

Let the equation of the variable line L be x a + y b = 1 . The line meets the x -axis at A(a, 0) and the y -axis at B(0, b) . The perpendicular distance from the origin (0,0) to the line L is given as 3 units. Using the perpendicular distance formula: |-1| 1 a^2 + 1 b^2 = 3 Squaring both sides, we get: 1 a^2 + 1 b^2 = 1 9 Let C(h, k) be the point that divides the line segment AB internally in the ratio 2 : 1 . This means AC : CB = 2 : 1 . Using the section formula, the coordinates of C are: h = 2(0) + 1(a) 2 + 1 =

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