The straight line through `P(x_1, y_1)` inclined at an angle `theta` with the x-axis meets the line `a x+b y+c=0\ in\ Qdot` Find the length of `P Qdot`

Explanation

The equation of straight line passing through the point (x_1,y_1) making an angle `theta` with x-axis is,

`frac{x−x_1}{costheta}=frac{y−y_1}{sintheta}=r`
where r is the distance of any point Q on this line from the point P`(x_1,y_1)`

Any point Q `=(x_1+rcostheta,y_1+rsintheta)` is on the line `ax+by+c=0`

`thereforea(x_1+rcostheta)+b(y_1+rsintheta)+c=0`

`\implies r=-frac{ax_1+by_1+c}{acostheta+bsintheta}`

`\implies `PQ`=|frac{ax_1+by_1+c}{acostheta+bsintheta}|`