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:
To find the area of the triangle formed by a line and the coordinate axes, we must first find the equation of the line and its x and y-intercepts.
Step 2: Key Formula or Approach:
Find the slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
Form the equation using \(y - y_1 = m(x - x_1)\).
Find the intercepts by setting \(x = 0\) (for y-intercept) and \(y = 0\) (for x-intercept).
The area of the right triangle is \(\frac{1}{2} |x_{\text{int}} \cdot y_{\text{int}}|\).
Step 3: Detailed Explanation:
Calculate the slope \(m\):
\[ m = \frac{-5 - 5}{3 - 1} = \frac{-10}{2} = -5 \] Equation of the line:
\[ y - 5 = -5(x - 1) \] \[ y = -5x + 5 + 5 \] \[ y = -5x + 10 \implies 5x + y = 10 \] Find the x-intercept (set \(y = 0\)):
\[ 5x = 10 \implies x = 2 \] The point is \(A(2, 0)\).
Find the y-intercept (set \(x = 0\)):
\[ y = 10 \] The point is \(B(0, 10)\).
Area of \(\Delta OAB\):
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 10 = 10 \] Step 4: Final Answer:
The area is 10.