Question:
<p>A straight line cuts off the intercepts OA = a and OB = b on the positive directions of x-axis and y-axis respectively. If the perpendicular from origin O to this line makes an angle of <span class="math-tex">\(\frac {\pi}6\)</span> with positive direction of y-axis and the area of <span class="math-tex">\(\triangle\)</span>OAB is <span class="math-tex">\(\frac{98}{3} \sqrt{3}\)</span>, then a<sup>2</sup> - b<sup>2</sup> is equal to:</p>
<p style="display:inline">98</p>
<p style="display:inline"><span class="math-tex">\(\frac{196}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{392}{3}\)</span></p>
<p style="display:inline">196</p>
Step-by-Step Solution
Key Concept: Use the intercept form of line (x/a + y/b = 1) and the condition that the perpendicular from origin makes angle π/6 with y-axis. The perpendicular distance formula p = ab/√(a² + b²) relates to the angle condition, and the area constraint ab/2 = 98√3/3 provides the second equation to solve for a² - b².
<p>Since, graph of given line is given by<br />
So, Equation of straight line: <span class="math-tex">\(\frac{x}{a}+\frac{y}{b}\)</span> = 1<br />
<img src="https://media-mycbseguide.s3.amazonaws.com/images/question_images/1700042456-bgav9q.jpg" style="height:133px; width:150px" /><br />
or x cos<span class="math-tex">\(\frac{\pi}{3}\)</span> + y sin<span class="math-tex">\(\frac{\pi}{3}\)</span> = p<br />
<span class="math-tex">\(\frac{x}{2}+\frac{y \sqrt{3}}{2}=p \)</span> <span class="math-tex">\(\Rightarrow \frac{x}{2 p}+\frac{y}{\frac{2 p}{\sqrt{3}}}\)</span> = 1<br />
Comparing both we get: a = 2p, b = <span class="math-tex">\(\frac{2 p}{\sqrt{3}}\)</span><br />
Now, area of <span class="math-tex">\(\triangle\)</span>OAB = <span class="math-tex">\(\frac{1}{2} \cdot a b=\frac{98}{3} \cdot \sqrt{3}\)</span><br />
<span class="math-tex">\(\Rightarrow \frac{1}{2} \cdot 2 p \cdot \frac{2 p }{\sqrt{3}}=\frac{98}{3} \cdot \sqrt{3}\)</span> <span class="math-tex">\( \Rightarrow\)</span> p<sup>2</sup> = 49<br />
Now, a<sup>2</sup> - b<sup>2</sup> = 4p<sup>2</sup> - <span class="math-tex">\(\frac{4 p ^2}{3}\)</span> <span class="math-tex">\(=\frac{2}{3} 4 p ^2=\frac{8}{3} \cdot 49=\frac{392}{3}\)</span></p>
Correct Answer: C