Question

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

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

Answer 1

Answer: (B)

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

Answer 2

If the given lines represent the same line, then the length of the perpendiculars from the origin to the lines are equal, so that

$\frac { c } { \sqrt { 1 + m ^ { 2 } } } =$ $\frac { p } { \sqrt { \cos ^ { 2 } \alpha + \sin ^ { 2 } \alpha } }$

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