Question

The lines $x\cos \alpha +y\sin \alpha =p_1$ and $x\cos \beta +y\sin \beta =p_2$ will be perpendicular, if

• A. $\alpha =\beta$
• B. $|\alpha -\beta |=\pi /2$
• C. $\alpha =\pi /2$
• D. $\alpha \pm \beta =\pi /2$

Answer 1

The correct option is B $|\alpha -\beta |=\pi /2$

$Slopeoffirstline{m}_1=\left(\frac{-cos\alpha }{sin\alpha }\right)Slopeofsecondline{m}_2=\left(\frac{-cos\beta }{sin\beta }\right)nowforlinestobe{\perp }^{r}toeachother{m}_1{m}_2=-1\left(\frac{-cos\alpha }{sin\alpha }\right)\times \left(\frac{-cos\beta }{sin\beta }\right)=-1orcos\alpha cos\beta +sin\alpha sin\beta =0cos(\alpha -\beta )=0orcos(\alpha -\beta )=cos\left(\frac{\pi }{2}\right)\therefore |\alpha -\beta |=\left(\frac{\pi }{2}\right)$