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

Question

If the equation y = mx + c and x cos α+ y sin α= p represent the same straight line, then (A) p = c √1+ m 2 (B) c=p √1+m2 (C) cp =√1+ m 2 (D) p 2+ c 2+ m 2=1

Tardigrade Admin · Accepted Answer

If the given lines represent the same line, then the lengths of the perpendiculars from the origin to the lines are equal, so that (c/√1+ m 2)=(p/√ cos 2 α+ sin 2 α) or c=p √1+m2

Q. If the equation $y = m x + c$ and $x cos α + y sin α = p$ represent the same straight line, then

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

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

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

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

If the given lines represent the same line, then the lengths of the perpendiculars from the origin to the lines are equal, so that
$\frac{c}{1 + m ^{2}} = \frac{p}{c o s ^{2} α + s i n ^{2} α}$
or $c = p 1 + m^{2}$

Q. If the equation y=mx+c and xcosα+ysinα=p represent the same straight line, then

p=c1+m2​

c=p1+m2​

cp=1+m2​

p2+c2+m2=1