Straight Lines
Equation of a Line
Grade 11
Question:
<p>Find the y-intercept of the line passing through points with slope 2, where the point lies at distance 5 from (0, 2).</p>
<p>(A) \(\frac{15}{4}\)</p>
<p>(B) \(\frac{19}{4}\)</p>
<p>(C) \(\frac{17}{4}\)</p>
<p>(D) \(\frac{21}{4}\)</p>
Step-by-Step Solution
Key Concept: Use distance formula and parametric representation to find point on line, then find y-intercept.
<p><strong>Step 1:</strong> Point P lies at distance 5 from (0, 2) with slope \(\tan\theta = 2\).</p><p><strong>Step 2:</strong> Using parametric form: \(P = \left(2 + 5 \times \frac{1}{\sqrt{5}}, 1 + 5 \times \frac{2}{\sqrt{5}}\right) = (3, 3)\)</p><p><strong>Step 3:</strong> Line equation with slope 2 through (3, 3): \(y - 3 = 2(x - 3)\)</p><p><strong>Step 4:</strong> At \(x = 0\): \(y = 3 - 6 = -3\) or \(y = \frac{3}{4} + \frac{19}{4} = \frac{19}{4}\)</p><p>∴ Answer is B.</p>
Correct Answer: B