If the straight line through the point P(3, 4) makes an angle π6 with the x-axis in the positive direction and meets the line 3x+5y+1=0 at Q, the length PQ is

Q=3+rCosπ6, 4+rSinπ6 =3+3r2, 4+r2 =33+3r2+54+r2+1=0 r =−30×235−52=3033−5 .

If the straight line through the point P(3, 4) makes an angle π6 with the x-axis in the positive direction and meets the line 3x+5y+1=0 at Q, the length PQ is

Q=3+rCosπ6, 4+rSinπ6 =3+3r2, 4+r2 =33+3r2+54+r2+1=0 r =−30×235−52=3033−5 .

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If the straight line through the point P(3, 4) makes an angle $\frac{π}{6}$ with the x-axis in the positive direction and meets the line $3 x + 5 y + 1 = 0$ at Q, the length PQ is

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$30 (3 \sqrt{3} - 5)$

$30$

$30 (\sqrt{3} - 1)$

$\sqrt{3} - 1$

answer is A.

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$Q = (3 + r C o s \frac{π}{6}, 4 + r S i n \frac{π}{6})$

$\begin{array}{l} = (3 + \frac{\sqrt{3} r}{2}, 4 + \frac{r}{2}) \\ = 3 (3 + \frac{\sqrt{3} r}{2}) + 5 (4 + \frac{r}{2}) + 1 = 0 \\ |r| = |\frac{- 30 \times 2 (3 \sqrt{5} - 5)}{2}| = 30 (3 \sqrt{3} - 5) . \end{array}$

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If the straight line through the point P(3, 4) makes an angle π6 with the x-axis in the positive direction and meets the line 3x+5y+1=0 at Q, the length PQ is

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If the straight line through the point P(3, 4) makes an angle $\frac{π}{6}$ with the x-axis in the positive direction and meets the line $3 x + 5 y + 1 = 0$ at Q, the length PQ is