Straight Lines
Grade None
Question:
<p>A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is</p>
<p style="display:inline">3x + 2y = xy</p>
<p style="display:inline">2x + 3y = xy</p>
<p style="display:inline">3x + 2y = 6xy</p>
<p style="display:inline">3x + 2y = 6</p>
Step-by-Step Solution
Key Concept: Relate the coordinates of vertex R to the intercepts of the line and use the given fixed point to establish the locus equation.
<p><img alt="" data-imgur-src="7F1e0KR.png" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/7F1e0KR.png" style="width: 200px; height: 117px;" /><br />
Equation of line PQ is <span class="math-tex">\(\frac{x}{\alpha}+\frac{y}{\beta}=1\)</span><br />
Since this line is passes through fixed point (2, 3).<br />
<span class="math-tex">\(\therefore \quad \frac{2}{\alpha}+\frac{3}{\beta}=1\)</span><br />
<span class="math-tex">\(\therefore\)</span> Locus of R is <span class="math-tex">\(2 \beta+3 \alpha=\alpha \beta\)</span><br />
i.e. 2y + 3x = xy <span class="math-tex">\(\Rightarrow\)</span> 3x + 2y = xy</p>
Correct Answer: A