Straight Lines
Grade None
Question:
<p>The distance of the point <span class="math-tex">\((2,3)\)</span> from the line <span class="math-tex">\(2 x-3 y+28=0\)</span>, measured parallel to the line <span class="math-tex">\(\sqrt{3 x}-y+1=0\)</span>, is equal to</p>
<p style="display:inline"><span class="math-tex">\(4 \sqrt{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(6 \sqrt{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(3+4 \sqrt{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(4+6 \sqrt{3}\)</span></p>
Step-by-Step Solution
Key Concept: The distance of a point from a line measured along a specific direction is obtained by dividing the perpendicular distance from the point to the line by the sine of the angle between the target line and the direction of measurement.
<p><img src="https://media-mycbseguide.s3.amazonaws.com/images/question_images/1757057605-n496r8.jpg" style="height:170px; width:250px" /><br />
Let 'Q' be the angle <span class="math-tex">\(b / w\)</span> two given lines<br />
<span class="math-tex">\(\tan \theta=\left|\frac{\frac{2}{3}-\sqrt{3}}{1+\frac{2}{3} \sqrt{3}}\right|\)</span><br />
<span class="math-tex">\(\tan \theta=\left|\frac{2-3 \sqrt{3}}{3+2 \sqrt{3}}\right|\)</span><br />
<span class="math-tex">\(\tan \theta=\frac{3 \sqrt{3}-2}{2 \sqrt{3}+3}\)</span><br />
Now<br />
<span class="math-tex">\({Pm} =\left|\frac{2(2)-3(3)+28}{\sqrt{4+9}}\right|\)</span><br />
<span class="math-tex">\(=\frac{23}{\sqrt{13}}\)</span><br />
<span class="math-tex">\(\because {PQ} =\frac{{Pm}}{\sin \theta}\)</span><br />
<span class="math-tex">\(=\operatorname{Pm} \operatorname{cosec} \theta\)</span><br />
<span class="math-tex">\(=\frac{23}{\sqrt{13}}\left(\frac{\sqrt{(2 \sqrt{3}+3)^{2}+(3 \sqrt{3}-2)^{2}}}{3 \sqrt{3}-2}\right)\)</span><br />
<span class="math-tex">\(=\frac{23}{\sqrt{13}}\left(\frac{\sqrt{12+9+12 \sqrt{3}+27+4-12 \sqrt{3}}}{3 \sqrt{3}-2}\right)\)</span><br />
<span class="math-tex">\(=\frac{23}{\sqrt{13}} \times \frac{\sqrt{52}}{(3 \sqrt{3}-2)}\)</span><br />
<span class="math-tex">\(=\frac{46}{3 \sqrt{3}-2} \times \frac{3 \sqrt{3}+2}{3 \sqrt{3}+2}\)</span><br />
<span class="math-tex">\(=\frac{46(3 \sqrt{3}+2)}{23}\)</span><br />
<span class="math-tex">\(=4+6 \sqrt{3}\)</span></p>
Correct Answer: D