<p>Find the value of the y-intercept \(c\) of the line \(y = 2x + c\) such that the perpendicular distance from the origin to this line equals 5.</p>
Step-by-Step Solution
Key Concept: Use the perpendicular distance formula from a point to a line to find the y-intercept.
<p><strong>Given:</strong> The line is \(y = 2x + c\), which can be written as \(-2x + y - c = 0\).</p><p><strong>Step 1:</strong> The perpendicular distance from origin to the line \(ax + by + c = 0\) is \(\frac{|c|}{\sqrt{a^2 + b^2}}\).</p><p><strong>Step 2:</strong> For the line \(-2x + y - c = 0\), the distance is \(\frac{|-c|}{\sqrt{(-2)^2 + 1^2}} = \frac{|c|}{\sqrt{5}}\).</p><p><strong>Step 3:</strong> Given that this distance equals 5: \(\frac{|c|}{\sqrt{5}} = 5\)</p><p><strong>Step 4:</strong> Solving: \(|c| = 5\sqrt{5}\)</p><p><strong>∴</strong> \(c = ±5\sqrt{5}\)</p>
Correct Answer: 5 or -5