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Question 6.

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

A
The distance between the y-intercepts of the two lines is 6
B
The distance from the origin to one of the lines is 32\dfrac{3}{\sqrt{2}}
C
The line L1L_1 passes through the point (1, 0)
D
The distance between the x-intercepts of the two lines is 4
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Text Explanation

Given that the equation of line L1L_1 is y=k(x−1)y = k(x - 1) and it intersects line L2L_2 at (5, 8).

So, the point (5, 8) must satisfy the equation of line L1L_1

We get 8=k(5−1)8=k\left(5-1\right)

or k=2k=2

Hence, the equation of line L1L_1 is y=2(x−1)y = 2(x - 1) . . . (1)

Now, L2L_2 has a slope of 1 Using slope form, the equation of line is y=mx+cy=mx+c, where mm is the slope and cc is the y-intercept

So, the equation of line L2L_2 is y=x+cy=x+c

Now, as it passes through the point (5, 8), the equation must satisfy

Hence, 8=5+c8=5+c

or, c=3c=3

Hence, the equation of line L2L_2 is y=x+3y=x+3 . . . (2)

Plotting both lines on the graph, we get

From the graph we can clearly see that the distance between the y-intercepts of the two lines is 5 not 6.

Hence, the statement the distance between the y-intercepts of the two lines is 6 is definitely false.

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