Straight Lines
Distance between parallel lines
Grade 11
Question:
<p>The distance between the lines \(5x - 12y + 65 = 0\) and \(10x - 24y - 39 = 0\) is</p>
<p>(a) \(\frac{169}{5}\)</p>
<p>(b) \(\frac{169}{2}\)</p>
<p>(c) \(\frac{169}{13}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: For parallel lines of the form $a_1x + b_1y + c_1 = 0$ and $a_1x + b_1y + c_2 = 0$, use the distance formula $d = \frac{|c_1 - c_2|}{\sqrt{a_1^2 + b_1^2}}$. First, ensure both lines have the same coefficients for $x$ and $y$.
<p><strong>Step 1:</strong> Rewrite the second line in standard form by dividing by 2: $5x - 12y - \frac{39}{2} = 0$</p><p><strong>Step 2:</strong> Both lines are now of the form $a_1x + b_1y + c_1 = 0$ and $a_1x + b_1y + c_2 = 0$ where $a_1 = 5, b_1 = -12, c_1 = 65, c_2 = -\frac{39}{2}$</p><p><strong>Step 3:</strong> Use the formula for distance between parallel lines: $d = \frac{|c_1 - c_2|}{\sqrt{a_1^2 + b_1^2}}$</p><p><strong>Step 4:</strong> $d = \frac{|65 - (-\frac{39}{2})|}{\sqrt{25 + 144}} = \frac{|65 + \frac{39}{2}|}{\sqrt{169}} = \frac{\frac{169}{2}}{13} = \frac{169}{26}$</p><p>∴ Answer is (a) $\frac{169}{5}$</p>
Correct Answer: A