Question:
<p>Two vertices of a triangle are (0, 2) and (4, 3). If its orthocentre is at the origin, then its third vertex lies in which quadrant?</p>
<p style="display:inline">first</p>
<p style="display:inline">third</p>
<p style="display:inline">second</p>
<p style="display:inline">fourth</p>
Step-by-Step Solution
Key Concept: The orthocenter is the point of intersection of altitudes, meaning the line segment connecting a vertex to the orthocenter is perpendicular to the side opposite that vertex.
<p>Since, m<sub>QR</sub> <span class="math-tex">\(\times\)</span> m<sub>PH</sub> = 1<br />
<span class="math-tex">\(\Rightarrow\)</span> m<sub>QR</sub> = <span class="math-tex">\(-\frac{1}{m_{P H}}\)</span><br />
<span class="math-tex">\(\Rightarrow\)</span> m<sub>QR</sub> = <span class="math-tex">\(\frac{y-3}{x-4}\)</span> = 0<br />
<span class="math-tex">\(\Rightarrow\)</span> y = 3<br />
m<sub>PQ</sub> <span class="math-tex">\(\times\)</span> m<sub>RH</sub> = -1<br />
<span class="math-tex">\(\Rightarrow \frac{1}{4} \times \frac{y}{x}\)</span> = -1<br />
<span class="math-tex">\(\Rightarrow\)</span> y = -4x<br />
<span class="math-tex">\(\Rightarrow\)</span> x = <span class="math-tex">\(-\frac{3}{4}\)</span><br />
Vertex R is <span class="math-tex">\(\left(\frac{-3}{4}, 3\right)\)</span><br />
Hence, vertex R lies in second quadrant.</p>
Correct Answer: C