Question

If two lines drawn from a point $P(2,3)$ intersect the line $x+y=6$ at a distance $\sqrt{\dfrac{2}{3}}$, then the angle between the lines is

Hint:

For such geometry problems, remember the standard angle formula using perpendicular distance.

Answer 1

Correct Answer: 1

Step 1: Use formula for angle between pair of lines from a point to a line.
Angle $\theta$ between two lines drawn from point $P$ to a line at perpendicular distance $d$ is given by \[ \sin\frac{\theta}{2}=\frac{d}{\text{distance from point to line}} \] Step 2: Find distance of point from the line.
Distance of $P(2,3)$ from $x+y-6=0$ is \[ D=\frac{|2+3-6|}{\sqrt{1^2+1^2}}=\frac{1}{\sqrt2} \] Step 3: Substitute values.
\[ \sin\frac{\theta}{2} =\frac{\sqrt{\frac{2}{3}}}{\frac{1}{\sqrt2}} =\sqrt{\frac{4}{3}} =\frac{2}{\sqrt3} \] \[ \Rightarrow \sin\frac{\theta}{2}=\frac{\sqrt3}{2} \] Step 4: Find $\theta$.
\[ \frac{\theta}{2}=\frac{\pi}{6} \Rightarrow \theta=\frac{\pi}{3} \] Final conclusion.
The angle between the two lines is $\dfrac{\pi}{3}$.