Question:
<p>Two sides of a rhombus are along the lines, x - y + 1 = 0 and 7x - y - 5 = 0. If its diagonals intersect at (-1, - 2), then which one of the following is a vertex of this rhombus?</p>
<p style="display:inline"><span class="math-tex">\(\left(\frac{1}{3},-\frac{8}{3}\right)\)</span></p>
<p style="display:inline">(-3, -9)</p>
<p style="display:inline">(-3, -8)</p>
<p style="display:inline"><span class="math-tex">\(\left(-\frac{10}{3},-\frac{7}{3}\right)\)</span></p>
Step-by-Step Solution
Key Concept: The diagonals of a rhombus bisect each other, making their intersection point the midpoint of both pairs of opposite vertices.
<p>As the given lines x - y + 1 = 0 and 7x - y - 5 = 0 are not parallel, therefore they represent the adjacent sides of the rhombus.<br />
On solving x - y + 1 = 0 and 7x - y - 5 = 0, we get x = 1 and y = 2. Thus, one of the vertex is A(1, 2).<img alt="" data-imgur-src="mvR7iUC.png" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/mvR7iUC.png" style="width: 150px; height: 116px;" /><br />
Let the coordinate of point C be (x, y)<br />
Then, <span class="math-tex">$-1=\frac{x+1}{2}$</span> and <span class="math-tex">$-2=\frac{y+2}{2}$</span><br />
<span class="math-tex">$\Rightarrow$</span> x + 1 = - 2 and y = - 4 - 2<br />
<span class="math-tex">$\Rightarrow$</span> x = - 3<br />
and y = - 6<br />
Hence, coordinates of C = (- 3, - 6)<br />
Note that, vertices B and D will satisfy x - y + 1 = 0 and 7x - y - 5 = 0, respectively<br />
Since, this option satisfies 7x - y - 5 = 0, therefore coordinate of vertex D is m<span class="math-tex">$\left(\frac{1}{3}, \frac{-8}{3}\right)$</span><br />
</p>
Correct Answer: A