Question

Consider the following lines in the $XY$-plane:
$L_1 : 5x - 2y = 1$,
$L_2 :$ the line passing through $(0, 1)$ and $(100, 101)$,
$L_3 :$ the line passing through $(1, 11)$ and parallel to the vector $-\hat{i} + 2\hat{j}$.
Let $A = (L_1 \cap L_2) \cup (L_2 \cap L_3) \cup (L_3 \cap L_1)$. What is the total number of elements of $A$?

• A. 3
• B. 0
• C. 1
• D. 2

Hint:

If the slopes of three lines are all distinct, they are not parallel.
They will form a triangle with exactly 3 intersection points unless they are concurrent.
Simply finding the intersection of two lines and verifying if it lies on the third is a highly efficient way to confirm non-concurrency.

Answer 1

The Correct Option is A

To determine the total number of elements in the set \( A = (L_1 \cap L_2) \cup (L_2 \cap L_3) \cup (L_3 \cap L_1) \), we need to find the intersection points of the lines \( L_1 \), \( L_2 \), and \( L_3 \).

Step 1: Equation of Line \( L_1 \)

The equation of line \( L_1 \) is given as:

\(5x - 2y = 1\)

This can be rewritten to find the slope-intercept form:

\(y = \frac{5}{2}x - \frac{1}{2}\)

Step 2: Equation of Line \( L_2 \)

Line \( L_2 \) passes through the points \( (0, 1) \) and \( (100, 101) \). The slope \( m_2 \) is:

\(m_2 = \frac{101 - 1}{100 - 0} = 1\)

The equation of the line passing through \( (0, 1) \) with a slope of 1 is:

\(y = x + 1\)

Step 3: Equation of Line \( L_3 \)

Line \( L_3 \) is parallel to the vector \(-\hat{i} + 2\hat{j}\), meaning the slope is:

\(m_3 = \frac{2}{-1} = -2\)

Since it passes through \( (1, 11) \), the equation is:

\(y - 11 = -2(x - 1)\)

Simplifying, we get:

\(y = -2x + 13\)

Step 4: Finding Intersection Points

1. Intersection of \( L_1 \) and \( L_2 \):

Equate the equations: \(\frac{5}{2}x - \frac{1}{2} = x + 1\)

Simplifying, we get:

\(x = \frac{3}{2}, \, y = \frac{5}{2}\)

1. Intersection of \( L_2 \) and \( L_3 \):

Equate the equations: \(x + 1 = -2x + 13\)

Simplifying, we get:

\(x = 4, \, y = 5\)

1. Intersection of \( L_3 \) and \( L_1 \):

Equate the equations: \(-2x + 13 = \frac{5}{2}x - \frac{1}{2}\)

Simplifying, we find:

\(x = 2, \, y = 9\)

Conclusion: Each pair of lines intersects at a unique point. Thus, the total number of distinct intersection points (elements of \( A \)) is 3.

The correct answer is: 3.