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

Answer 1

The correct option is B

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

Here $(AB{)}^2=(a-c{)}^2+(b-d{)}^2(OA{)}^2=(a-0{)}^2+(b-0{)}^2=a^2+b^2\text{and}(OB{)}^2=c^2+d^2$


$\text{In}\Delta AOB,\cos \theta =\frac{(OA{)}^2+(OB{)}^2-(AB{)}^2}{2OA.OB}=\frac{a^2+b^2+c^2+d^2-(a-c{)}^2+(b-d{)}^2}{2\sqrt{a^2+b^2}.\sqrt{c^2+d^2}}\frac{ac+bd}{\sqrt{(a^2+b^2)(c^2+d^2)}}.$