Question:
<p>ABCD is a cyclic quadrilateral, where AB, BC, CD are represented by x + y - 1 = 0, 2x + y + 1 = 0 and x - 3y + 5 = 0 respectively. If DA passes through (2, 1), then the equation of DA is</p>
<p style="display:inline">4x + 3y - 11 = 0</p>
<p style="display:inline">3x - 4y - 2 = 0</p>
<p style="display:inline">3x + 4y - 10 = 0</p>
<p style="display:inline">4x - 3y - 5 = 0</p>
Step-by-Step Solution
Key Concept: For a cyclic quadrilateral ABCD, the four sides lie on a circle. Use the family of circles passing through the intersection points of three given lines (AB, BC, CD) to find the fourth line DA. The equation of the circle is expressed as a linear combination of the three line equations, then DA is found using the constraint that it passes through (2,1) and completes the cyclic quadrilateral.
<html><body><p><img alt="" data-imgur-src="Zxear1x.png" height="131" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1623761597-78a56p.jpg" width="232"/><br/>
Circle through A, B, C, D is given by<br/>
(mx - y + c) (2x + y + 1) + <span class="math-tex">$\lambda$</span>(x + y - 1)<span class="math-tex">$\cdot$</span> (x - 3y + 5) = 0<br/>
<span class="math-tex">$\Rightarrow$</span> coefficient of x<sup>2</sup> = coefficient of y<sup>2</sup><br/>
<span class="math-tex">$\Rightarrow$</span> 2m + <span class="math-tex">$\lambda$</span> = -1 - 3<span class="math-tex">$\lambda$</span> <span class="math-tex">$\Rightarrow \lambda=\frac{-(1+2 m)}{4}$</span> ...(i)<br/>
Also, coefficient of (xy) = 0<br/>
<span class="math-tex">$\Leftrightarrow$</span> m - 2 - 2<span class="math-tex">$\lambda$</span> = 0 <span class="math-tex">$\Rightarrow \lambda=\frac{m-2}{2}$</span> ...(ii)<br/>
From (i) and (ii), we get<br/>
<span class="math-tex">$\frac{-(1+2 m)}{4}=\frac{m-2}{2}$</span><br/>
<span class="math-tex">$\Leftrightarrow$</span> 2m - 4 = -1 - 2m<br/>
<span class="math-tex">$\Leftrightarrow m=\frac{3}{4}$</span><br/>
The equation of DA passing through (2, 1) is 3x - 4y - 2 = 0</p></body></html>
Correct Answer: B