Question:
<p>Chord of the curve 4x<sup>2</sup> + y<sup>2</sup> - x + 4y = 0 which subtend a right angle at the origin pass through a fixed point whose co-ordinate are :</p>
<p style="display:inline"><span class="math-tex">\(\left(\frac{1}{5}, \frac{-4}{5}\right)\)</span></p>
<p style="display:inline"><span class="math-tex">\(\left(\frac{-1}{5}, \frac{-4}{5}\right)\)</span></p>
<p style="display:inline"><span class="math-tex">\(\left(\frac{1}{5} , \frac{4}{5}\right)\)</span></p>
<p style="display:inline"><span class="math-tex">\(\left(\frac{-1}{5}, \frac{4}{5}\right)\)</span></p>
Step-by-Step Solution
Key Concept: Homogenize the curve's equation using the linear chord equation to represent the pair of lines passing through the origin, then apply the condition that the sum of the coefficients of x² and y² must be zero for a right angle.
<p><img alt="" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1619691798-9vv4jb.jpg" style="height:85px; width:100px" /><br />
Let chord be ax + by = 1<br />
homogenising cube with the help of line<br />
4x<sup>2</sup> + y<sup>2</sup> - x (ax + by) + 4y (ax + by) = 0<br />
Subtending right angle<br />
4 - a + 1 + 4b = 0<br />
a - 4b = 5<br />
<span class="math-tex">$\frac{a}{5}-\frac{4}{5}$</span> b = 1<br />
passing through <span class="math-tex">$\left(\frac{1}{5}, \frac{-4}{5}\right)$</span></p>
Correct Answer: A