Question

A straight line L passes through (2,8) and the origin. Find the equation of a line perpendicular to L.
1. The line passes through the origin.
2. The line passes through (2,-0.5).

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question ask.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hint:

Always start by extracting all possible information from the question prompt itself before looking at the statements. Here, the slope of the perpendicular line was found from the prompt, which simplified the analysis of the statements.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question asks for the equation of a line that is perpendicular to a given line L. To find a unique equation for a line, we need its slope and a point on the line.
Step 2: Key Formula or Approach:
1. Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
2. Perpendicular slopes: If a line has slope \( m \), a perpendicular line has a slope of \( m_{\perp} = -\frac{1}{m} \).
3. Point-slope form of a line: \( y - y_1 = m(x - x_1) \).
Step 3: Detailed Explanation:
First, let's use the information in the question prompt to find the slope of line L.
Line L passes through the origin (0, 0) and (2, 8).
Slope of L, \( m_L = \frac{8 - 0}{2 - 0} = \frac{8}{2} = 4 \).
The line we are looking for is perpendicular to L. Let its slope be \( m_{\perp} \).
\( m_{\perp} = -\frac{1}{m_L} = -\frac{1}{4} \).
So, we know the slope of the perpendicular line is \( -1/4 \). Its equation is of the form \( y = -\frac{1}{4}x + c \). To find the full equation, we need to find the y-intercept, c, which requires a point on this perpendicular line.
Analyze Statement (1): "The line passes through the origin."
This refers to the perpendicular line. If the perpendicular line passes through the origin (0, 0), we can find c.
Using the point (0, 0) in \( y = -\frac{1}{4}x + c \):
\[ 0 = -\frac{1}{4}(0) + c \implies c = 0 \] The equation of the line is \( y = -\frac{1}{4}x \). This is a unique equation. Thus, Statement (1) is sufficient.
Analyze Statement (2): "The line passes through (2,-0.5)."
This also refers to the perpendicular line. We use the point (2, -0.5) to find c.
\[ y = -\frac{1}{4}x + c \] \[ -0.5 = -\frac{1}{4}(2) + c \] \[ -0.5 = -0.5 + c \implies c = 0 \] The equation of the line is again \( y = -\frac{1}{4}x \). This is a unique equation. Thus, Statement (2) is sufficient.
Step 4: Final Answer:
Since each statement alone provides enough information to determine the unique equation of the perpendicular line, the correct option is (D).