Question

The straight line passing through the points $(1, 5)$ and $(3, -5)$ meets the coordinate axes at the points $A$ and $B$. Then the area of the triangle $\triangle OAB$, where $O$ is the origin, is

• A. 2
• B. 4
• C. 5
• D. 8
• E. 10

Hint:

Converting the line equation to the form \( \frac{x}{a} + \frac{y}{b} = 1 \) is the fastest way to find the base and height of the triangle formed with the coordinate axes.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
First, we find the equation of the line. Then we find its $x$-intercept (where $y=0$) and $y$-intercept (where $x=0$). The area of the triangle formed by the origin and the axes is \( \frac{1}{2} |x\text{-intercept} \times y\text{-intercept}| \).
Key Formula or Approach:
1. Two-point form: \( y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \).
2. Area of triangle = \( \frac{1}{2} |ab| \), where $a$ and $b$ are the intercepts.

Step 2: Detailed Explanation:
Points: \( (1, 5) \) and \( (3, -5) \).
Slope \( m = \frac{-5 - 5}{3 - 1} = \frac{-10}{2} = -5 \).
Equation of the line:
\[ y - 5 = -5(x - 1) \]
\[ y - 5 = -5x + 5 \]
\[ 5x + y = 10 \]
Divide by 10 to get the intercept form \( \frac{x}{a} + \frac{y}{b} = 1 \):
\[ \frac{5x}{10} + \frac{y}{10} = 1 \implies \frac{x}{2} + \frac{y}{10} = 1 \]
The intercepts are \( a = 2 \) and \( b = 10 \).
Points are \( A(2, 0) \) and \( B(0, 10) \).
Area of \( \triangle OAB = \frac{1}{2} \times |2 \times 10| = \frac{1}{2} \times 20 = 10 \).

Step 3: Final Answer:
The area of the triangle is 10.