AP EAMCET · Maths · Straight Lines
If a straight line passing through the point \(\mathrm{P}(3,4)\) makes an angle \(\frac{\pi}{6}\) with the \(\mathrm{X}\)-axis and meets the line \(12 x+5 y+10=0\) at \(\mathrm{Q}\), then the length of \(\mathrm{PQ}\) is
- A \(\frac{132}{12 \sqrt{3}+5}\)
- B \(\frac{166}{8 \sqrt{3}+6}\)
- C \(\frac{182}{6 \sqrt{3}+4}\)
- D \(\frac{192}{14 \sqrt{3}+6}\)
Answer & Solution
Correct Answer
(A) \(\frac{132}{12 \sqrt{3}+5}\)
Step-by-step Solution
Detailed explanation
\( Q = (3 + r \cos(\frac{\pi}{6}), 4 + r \sin(\frac{\pi}{6})) = (3 + r \frac{\sqrt{3}}{2}, 4 + r \frac{1}{2}) \) \( 12(3 + r \frac{\sqrt{3}}{2}) + 5(4 + r \frac{1}{2}) + 10 = 0 \) \( 36 + 6\sqrt{3}r + 20 + \frac{5}{2}r + 10 = 0 \) \( 66 + (\frac{12\sqrt{3}+5}{2})r = 0 \)…
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