Question

If the equation $y = mx + c$ and $x \cos \alpha+ y \sin \alpha= p$ represent the same straight line, then

• A. $p = c \sqrt{1+ m ^{2}}$
• B. $c=p \sqrt{1+m^{2}}$
• C. $cp =\sqrt{1+ m ^{2}}$
• D. $p ^{2}+ c ^{2}+ m ^{2}=1$

Answer 1

Answer: (B)

If the given lines represent the same line, then the lengths of the perpendiculars from the origin to the lines are equal, so that
$\frac{c}{\sqrt{1+ m ^{2}}}=\frac{p}{\sqrt{\cos ^{2} \alpha+\sin ^{2} \alpha}}$
or $c=p \sqrt{1+m^{2}}$