Question:
<p>If the straight line, 2x - 3y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, <span class="math-tex">\(\beta\)</span>), then <span class="math-tex">\(\beta\)</span> equals</p>
<p style="display:inline">5</p>
<p style="display:inline">-5</p>
<p style="display:inline"><span class="math-tex">\(\frac{35}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(-\frac{35}{3}\)</span></p>
Step-by-Step Solution
Key Concept: Two non-vertical lines are perpendicular if and only if the product of their slopes is -1.
<p>Slope of the line 2x - 3y + 17 = 0 is <span class="math-tex">$\frac{2}{3}=m_{1}$</span> (let) and the slope of line joining the points (7, 17) and (15, <span class="math-tex">$\beta$</span>) is <span class="math-tex">$\frac{\beta-17}{15-7}=\frac{\beta-17}{8}$</span> = m<sub>2 </sub>(let)<br />
According to the question, m<sub>1</sub>m<sub>2</sub> = - 1<br />
<span class="math-tex">$\Rightarrow \frac{2}{3} \times \frac{\beta-17}{8}$</span> = -1 <span class="math-tex">$\Rightarrow$</span> <span class="math-tex">$\beta$</span> - 17 = -12 <span class="math-tex">$\Rightarrow$</span> <span class="math-tex">$\beta$</span> = 5</p>
Correct Answer: A