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Properties of Intersecting Lines

XAT 2025 · Quantitative Ability & Data Interpretation · Medium · Coordinate Geometry

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A straight line L1L_1 has the equation y=k(x−1)y = k(x - 1), where kk is some real number. The straight line L1L_1 intersects another straight line L2L_2 at the point (5,8)(5, 8). If L2L_2 has a slope of 1, which of the following is definitely FALSE?

  1. A.

    The distance between the y-intercepts of the two lines is 6

  2. B.

    The distance from the origin to one of the lines is 3/\sqrt{2}

  3. C.

    The line L_1 passes through the point (1, 0)

  4. D.

    The distance between the x-intercepts of the two lines is 4

  5. E.

    The distance from the origin to one of the lines is 2/\sqrt{5}

Answer

A

Explanation

Line L1L_1 passes through (5,8)  ⟹  8=k(5−1)  ⟹  k=2(5,8) \implies 8 = k(5 - 1) \implies k = 2. Equation of L1L_1: y=2(x−1)  ⟹  2x−y−2=0y = 2(x - 1) \implies 2x - y - 2 = 0. Line L2L_2 has slope 1 and passes through (5,8)  ⟹  y−8=1(x−5)  ⟹  y=x+3  ⟹  x−y+3=0(5,8) \implies y - 8 = 1(x - 5) \implies y = x + 3 \implies x - y + 3 = 0.

y-intercept of L1L_1 is (0,−2)(0, -2) and y-intercept of L2L_2 is (0,3)(0, 3). Distance between y-intercepts = 3−(−2)=53 - (-2) = 5. Therefore, the statement claiming the distance between the y-intercepts is 6 is definitely FALSE.

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