Question

A straight line \( L_1 \) has the equation \( y = k(x - 1) \), where \( k \) is some real number. The straight line \( L_1 \) intersects another straight line \( L_2 \) at the point (5, 8). If \( L_2 \) has a slope of 1, which of the following is definitely FALSE?

• A. The distance from the origin to one of the lines is \( \frac{3}{\sqrt{2}} \)
• B. The distance between the x-intercepts of the two lines is 4
• C. The distance between the y-intercepts of the two lines is 6
• D. The line \( L_1 \) passes through the point (1, 0)
• E. The distance from the origin to one of the lines is \( \frac{2}{\sqrt{5}} \)

Hint:

When analyzing geometrical problems involving lines, make sure to verify all conditions, such as slope, intercepts, and distances, to spot inconsistencies.

Answer 1

The Correct Option is A

Step 1: Evaluate the equation of \( L_1 \).
The equation of the line \( L_1 \) is given as \( y = k(x - 1) \), where \( k \) is the slope.
Step 2: Analyze the given options.
By plugging in the given slope and point of intersection, we can verify which options are correct and find that option (A) is the one that does not hold.
Final Answer: \[ \boxed{\text{(A) The distance from the origin to one of the lines is } \frac{3}{\sqrt{2}}} \]