Question:
<p>Axis of a parabola lies along X-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive X-axis, then which of the following points does not lie on it?</p>
<p style="display:inline">(4, -4)</p>
<p style="display:inline">(6, 4<span class="math-tex">\(\sqrt2\)</span>)</p>
<p style="display:inline">(8, 6)</p>
<p style="display:inline">(5, <span class="math-tex">\(2\sqrt6\)</span>)</p>
Step-by-Step Solution
Key Concept: Construct the parabola's equation using the vertex-shift form $(y-k)^2 = 4a(x-h)$, where the focal length $a$ is the distance between the vertex and the focus.
<p>According to given information, we have the following figure.<br />
<img alt="" data-imgur-src="KmJXTcU.png" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/KmJXTcU.png" style="width: 200px; height: 135px;" /><br />
Now, if the origin is shifted to (2, 0) and (X, Y) are the<br />
coordinates with respect to new origin, then equation of<br />
parabola is Y<sup>2</sup> = 4aX,<br />
where, X = x - 2 and Y = y and a = 4 - 2 = 2<br />
<span class="math-tex">$\therefore$</span> y<sup>2</sup> = 8(x - 2)<br />
Note that (8, 6) is the only point which does not satisfy the equation.</p>
Correct Answer: C