Question

If the equations $y=mx+c$ 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)

$c=p\sqrt{1+m^2}$

$y=mx+c$
$y-mx=c$
$\frac{y}{\sqrt{1+m^2}}-\frac{mx}{\sqrt{1+m^2}}=\frac{c}{\sqrt{1+m^2}}$
Comparing coefficients with $x\cos \alpha +y\sin \alpha =p$ we get
$\sin \alpha =\frac{1}{\sqrt{1+m^2}}$
$\cos \alpha =\frac{-m}{\sqrt{1+m^2}}$
$\frac{c}{\sqrt{1+m^2}}=p$
Therefore
$c=p\sqrt{1+m^2}$