Question:
<p>The normal drawn at P (-1, 2) on the circle x<sup>2</sup> + y<sup>2</sup> - 2x - 2y - 3 = 0 meets the circle at another point Q. Then the coordinates of Q are</p>
<p style="display:inline">(2, 0)</p>
<p style="display:inline">(-3, 0)</p>
<p style="display:inline">(-2, 0)</p>
<p style="display:inline">(3, 0)</p>
Step-by-Step Solution
Key Concept: The normal to a circle always passes through its center, so any normal line segment between two points on the circle is a diameter.
<p>Let the coordinates of Q be (x, y).<br />
Centre of the given circle is C = (1, 1)<br />
C = midpoint of PQ<br />
<span class="math-tex">\(\Rightarrow(1,1)=\left(\frac{x-1}{2}, \frac{y+2}{2}\right)\)</span><br />
<span class="math-tex">\(\Rightarrow \frac{x-1}{2}=1\)</span> and <span class="math-tex">\(\frac{y+2}{2}=1\)</span><br />
<span class="math-tex">\(\Rightarrow\)</span> x = 3 and y = 0<br />
<span class="math-tex">\(\therefore\)</span> Q = (3, 0)</p>
Correct Answer: D