Question

If the line segment joining the points $A(a,b)$ and $B(c,d)$ subtends an angle $\theta$ at the origin, then $\cos \theta$ is equal to

• A. $\frac{ab+cd}{\sqrt{(a^2+b^2)(c^2+d^2)}}$
• B. $\frac{ab+bd}{\sqrt{(a^2+b^2)(c^2+d^2)}}$
• C. $\frac{ac+bd}{\sqrt{(a^2+b^2)(c^2+d^2)}}$
• D. None of these

Answer 1

Answer: (C)

$\frac{ac+bd}{\sqrt{(a^2+b^2)(c^2+d^2)}}$

The equation of the line through the origin and $(a,b)$ is $y=\frac{b}{a}x$......(1)

And the equation of the line through the origin and $(c,d)$ is $y=\frac{d}{c}x$.......(2).

Given $\theta$ be the angle between (1) and (2).

Then

$\tan \theta =\frac{\frac{d}{c}-\frac{b}{a}}{1+\frac{bd}{ac}}$

or, $\tan \theta =\frac{ad-bc}{ac+bd}$

Then $\cos \theta =\frac{ac+bd}{\sqrt{(ad-bc{)}^2+(ac+bd{)}^2}}$

or, $\cos \theta =\frac{ac+bd}{\sqrt{a^2{d}^2+b^2{c}^2+a^2{c}^2+b^2{d}^2}}$

or, $\cos \theta =\frac{ac+bd}{\sqrt{(a^2+b^2)(c^2+d^2)}}$.