Straight Lines
Area using Coordinates
Grade 11
Question:
<p>The point \((2a, a)\) lies on the line \(2x + 3y = 20\). The area of the triangle formed by the point \((a, a)\), origin, and point on the line is what?</p>
<p>(A) \(\frac{200}{49}\)</p>
<p>(B) \(\frac{300}{49}\)</p>
<p>(C) \(\frac{400}{49}\)</p>
<p>(D) \(\frac{500}{49}\)</p>
Step-by-Step Solution
Key Concept: Find the value of parameter using the line equation, then compute triangle area using coordinate geometry.
<p><strong>Step 1:</strong> Point \((2a, a)\) lies on \(2x + 3y = 20\)</p><p><strong>Step 2:</strong> Substituting: \(4a + 3a = 20 \Rightarrow a = \frac{20}{7}\)</p><p><strong>Step 3:</strong> Triangle vertices: \(A(0, 0)\), \(B(a, 0) = \left(\frac{20}{7}, 0\right)\), \(C(2a, a) = \left(\frac{40}{7}, \frac{20}{7}\right)\)</p><p><strong>Step 4:</strong> Area \(= \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times \frac{20}{7} \times \frac{20}{7} = \frac{400}{49}\)</p><p>∴ Answer is C.</p>
Correct Answer: C