Question

Consider the following statements?
(1) The length $p$ of the perpendicular from the origin to the lines $ax+by=c$ satisfies the relation $p^2=\frac{c^2}{a^2+b^2}$
(2) The length $p$ of the perpendicular from the origin to the line $\frac{x}{a}+\frac{y}{b}=1$ satisfies the relation $\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$
(3) The length $p$ of the perpendicular from the origin to the line $y=mx+c$ satisfies the relation $\frac{1}{p^2}=\frac{1+m^2+c^2}{c^2}$
Which of the above is/are correct?

• A. 1, 2 and 4
• B. 1 only
• C. 1 and 2 only
• D. 2 only

Answer 1

The correct option is C 1 and 2 only

(1) Perpendicular from origin to line $az+by=c$

$\to P=\frac{c}{\sqrt{a^2+b^2}}\to P^2=\frac{c^2}{a^2+b^2}$

(2) Perpendicular from origin to line $[\frac{x}{a}+\frac{y}{b}=1]$

$P=\frac{1}{\sqrt{{\left(\frac{1}{a}\right)}^2+{\left(\frac{1}{b}\right)}^2}}\to \frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$

(3) $y=mx+c\to y-mx=c,P=\frac{c}{\sqrt{1^2+m^2}}$

Ans. C $1$ and $2$ only.