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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 32\frac{3}{\sqrt{2}}

  3. C.

    The line L1L_1 passes through the point (1,0)(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 25\frac{2}{\sqrt{5}}

Answer

A

Explanation

Since L1L_1 passes through (5,8)(5,8), 8=k(5−1)  ⟹  4k=8  ⟹  k=28 = k(5-1) \implies 4k = 8 \implies k = 2. So line L1L_1 is y=2(x−1)=2x−2y = 2(x - 1) = 2x - 2. Line L2L_2 has slope 1 and passes through (5,8)(5,8), so y−8=1(x−5)  ⟹  y=x+3y - 8 = 1(x - 5) \implies y = x + 3.

  • yy-intercept of L1L_1 is (0,−2)(0, -2).
  • yy-intercept of L2L_2 is (0,3)(0, 3). Distance between y-intercepts = 3−(−2)=53 - (-2) = 5. Therefore, option A ("The distance between the y-intercepts of the two lines is 6") is definitely FALSE.

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