If the straight line through the point (3,4) makes an angle `pi/6` with x-axis and meets the line 12 x + 5 y + 10 = 0 at Q, Then the length of PQ is :

Explanation

The equation of straight line passing through the point (3,4) making an angle `pi/6` with x-axis is,

`frac{x−3}{cos30^@}=frac{y−4}{sin30^@}=r`
where r is the distance of any point Q on this line from the point P`(3,4)`

Any point Q `=(frac{rsqrt3}{2}+3,r/2+4)` is on the line `12x+5y+10=0`

`therefore12(frac{rsqrt3}{2}+3)+5(r/2+4)+10=0`

`\implies(12sqrt3+5)r+132=0`

`\implies r=frac{|-132|}{12sqrt(3)+5}`

`=frac{132}{12sqrt(3)+5}`