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JEE MainMathematicsStraight Lines

Let A(2, 4) , B(0, -2) , and C(6, 0) be the vertices of a triangle ABC . A point P lies on the line segment BC . If the area of the region enclosed by the lines AP , AC , and the x -axis is 8 , then the ratio of the area of triangle APB to the area of triangle ABC is 1:k , where the value of k is

Options

  1. A3
  2. B2
  3. C4
  4. D5

Correct answer

A. 3

Step-by-step solution

Let the x -intercept of the line AC be R . The equation of the line AC passing through A(2, 4) and C(6, 0) is: y - 0 = 4 - 0 2 - 6 (x - 6) x + y = 6 The x -intercept of AC is R(6, 0) . Let the x -intercept of the line AP be Q(q, 0) . The area of the region enclosed by AP , AC , and the x -axis is the area of AQR . Area = 1 2 QR ( height of A) = 8 1 2 |q - 6| 4 = 8 |q - 6| = 4 q = 10 or q = 2 . Case 1: q = 10 . The line AP passes through A(2, 4) and Q(10, 0) . Equation of AP : x + 2y = 10 . Equation of BC passing th

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