Question:
<p>If (<span class="math-tex">\(\alpha\)</span>, <span class="math-tex">\(\beta\)</span>) is the orthocentre of the triangle ABC with vertices A(3, -7), B(-1, 2) and C(4, 5), then 9<span class="math-tex">\(\alpha\)</span> - 6<span class="math-tex">\(\beta\)</span> + 60 is equal to:</p>
<p style="display:inline">30</p>
<p style="display:inline">35</p>
<p style="display:inline">25</p>
<p style="display:inline">40</p>
Step-by-Step Solution
Key Concept: The orthocenter is the point of intersection of altitudes, found by solving the equations of lines passing through vertices perpendicular to their opposite sides.
<p><img src="https://media-mycbseguide.s3.amazonaws.com/images/question_images/1700731277-xswbz7.jpg" style="height:157px; width:150px" /><br />
Equation of AD: y + 7 = <span class="math-tex">$\frac {-5}3$</span>(x - 3) <span class="math-tex">$\Rightarrow$</span> 5x + 3y + 6 = 0<br />
Equation of BE: y - 2 = <span class="math-tex">$\frac{-1}{12}$</span>(x - 1)<br />
x + 12y = 23 <span class="math-tex">$\Rightarrow \alpha=\frac{-47}{19}, \beta=\frac{121}{57}$</span><br />
So, 9<span class="math-tex">$\alpha$</span> - 6<span class="math-tex">$\beta$</span> + 60 = 25</p>
Correct Answer: C