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

To solve this problem, we need to understand the relationships between the given lines, their equations, and their intersections. Let's break it down step by step.

Step 1: Understanding the Equations

• The line \(L_1\) has the equation \(y = k(x - 1)\). 
• The line \(L_2\) has a slope of 1, therefore, its equation can be represented as \(y = x + c\), where \(c\) is a constant.

Step 2: Intersection Point

It is given that the lines \(L_1\) and \(L_2\) intersect at the point (5, 8).

• Substituting (5, 8) in \(L_1\): \(8 = k(5 - 1) \Rightarrow 8 = 4k \Rightarrow k = 2\).
• Thus, \(L_1\) is \(y = 2(x - 1) = 2x - 2\).
• Substituting (5, 8) in \(L_2\): \(8 = 5 + c \Rightarrow c = 3\).
• Thus, \(L_2\) is \(y = x + 3\).

Step 3: Analyzing Each Option

1. The distance from the origin to one of the lines is \(\frac{3}{\sqrt{2}}\):
   • For \(L_1\): The distance from the origin (0,0) is given by \(\frac{|0 - 0 - 2|}{\sqrt{2^2 + (-1)^2}} = \frac{2}{\sqrt{5}}\).
   • For \(L_2\): The distance from the origin (0,0) is given by \(\frac{|0 - 0 - 3|}{\sqrt{1^2 + (-1)^2}} = \frac{3}{\sqrt{2}}\).
   • The statement is true for \(L_2\), but checking options asks to identify the FALSE outcome.
2. The distance between the x-intercepts of the two lines is 4:
   • X-intercept of \(L_1\): Set \(y = 0\), then \(0 = 2x - 2 \Rightarrow x = 1\).
   • X-intercept of \(L_2\): Set \(y = 0\), then \(0 = x + 3 \Rightarrow x = -3\).
   • Distance between x-intercepts: \(1 - (-3) = 4\).
   • This statement is true.
3. The distance between the y-intercepts of the two lines is 6:
   • Y-intercept of \(L_1\): Set \(x = 0\), then \(y = -2\).
   • Y-intercept of \(L_2\): Set \(x = 0\), then \(y = 3\).
   • Distance between y-intercepts: \(3 - (-2) = 5\).
   • This statement is false, as the actual distance is 5, not 6.
4. The line \(L_1\) passes through the point (1, 0):
   • Substitute (1,0) in \(L_1\): \(0 = 2(1) - 2 \Rightarrow 0 = 0\).
   • This point lies on the line, so this statement is true.
5. The distance from the origin to one of the lines is \(\frac{2}{\sqrt{5}}\):
   • This was already covered and is true for \(L_1\).

Conclusion

The statement that is definitely FALSE is: \("The distance between the y-intercepts of the two lines is 6."\) The actual distance is 5.

Answer 2

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: \[ \boxed{\text{(A) The distance from the origin to one of the lines is } \frac{3}{\sqrt{2}}} \]