Question:
<p>A chord AB drawn from the point A(0, 3) on circle x<sup>2</sup> + 4x +(y - 3)<sup>2</sup> = 0 meets to M in such a way that AM = 2AB, then the locus of point M will be</p>
<p style="display:inline">Hyperbola</p>
<p style="display:inline">Parabola</p>
<p style="display:inline">Circle</p>
<p style="display:inline">Straight line</p>
Step-by-Step Solution
Key Concept: Express the coordinates of the point lying on the circle in terms of the locus point using the section formula and substitute them into the given circle equation.
<p>Let M be (h, k).<br />
<span class="math-tex">\(\therefore\)</span> Point B is <span class="math-tex">\(\left(\frac{h}{2}, \frac{k+3}{2}\right)\)</span><br />
It lies on the given circle.<br />
<img alt="" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1623761557-gtx2pm.jpg" style="height:97px; width:116px" /><br />
<span class="math-tex">\(\therefore \left(\frac{h}{2}\right)^{2}+4\left(\frac{h}{2}\right)+\left(\frac{k+3}{2}-3\right)^{2}=0\)</span><br />
<span class="math-tex">\(\Rightarrow \frac{h^{2}}{4}+\frac{8 h}{4}+\frac{(k-3)^{2}}{4}=0\)</span><br />
<span class="math-tex">\(\Rightarrow\)</span> h<sup>2</sup> + 8h + k<sup>2</sup> - 6k + 9 = 0<br />
<span class="math-tex">\(\Rightarrow\)</span> x<sup>2</sup> + y<sup>2</sup> + 8x - 6y + 9 = 0, which is a circle.</p>
Correct Answer: C