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Consider the following statements:
1) The length $\mathrm{p}$ of the perpendicular from the origin to
the line $a x+b y=c$ satisfies the relation $p^{2}=\frac{c^{2}}{a^{2}+b^{2}}$.
2) The length $\mathrm{p}$ of the perpendicular from the origin to
the line $\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1$ satisfies the relation $\frac{1}{\mathrm{p}^{2}}=\frac{1}{\mathrm{a}^{2}}+\frac{1}{\mathrm{~b}^{2}}$.
3) The length $\mathrm{p}$ of the perpendicular from the origin to the line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ satisfies the relation $\frac{1}{\mathrm{p}^{2}}=\frac{1+\mathrm{m}^{2}+\mathrm{c}^{2}}{\mathrm{c}^{2}}$
Which of the above is/are correct?
Options:
1) The length $\mathrm{p}$ of the perpendicular from the origin to
the line $a x+b y=c$ satisfies the relation $p^{2}=\frac{c^{2}}{a^{2}+b^{2}}$.
2) The length $\mathrm{p}$ of the perpendicular from the origin to
the line $\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1$ satisfies the relation $\frac{1}{\mathrm{p}^{2}}=\frac{1}{\mathrm{a}^{2}}+\frac{1}{\mathrm{~b}^{2}}$.
3) The length $\mathrm{p}$ of the perpendicular from the origin to the line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ satisfies the relation $\frac{1}{\mathrm{p}^{2}}=\frac{1+\mathrm{m}^{2}+\mathrm{c}^{2}}{\mathrm{c}^{2}}$
Which of the above is/are correct?
Solution:
2993 Upvotes
Verified Answer
The correct answer is:
1 and 2 only
1: We know, the perpendicular distance from
$\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ to line $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$ is $\frac{\mid \mathrm{ax}_{1}+\mathrm{by}_{1}+\mathrm{c}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}$
Here, $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)=(0,0)$ and distance $=\mathrm{P}$.
$P=\frac{|a(0)+b(0)+(c)|}{\sqrt{a^{2}+b^{2}}}$
$\Rightarrow \mathrm{P}^{2}=\frac{\mathrm{c}^{2}}{\mathrm{a}^{2}+\mathrm{b}^{2}}$
1 is correct.
2: Line is $\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1 \Rightarrow\left(\frac{1}{\mathrm{a}}\right) \mathrm{x}+\left(\frac{1}{\mathrm{~b}}\right) \mathrm{y}+(-1)=0$
$P=\frac{\left|\frac{1}{a}(0)+\frac{1}{b}(0)-1\right|}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}}=\frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}}$
$=P^{2}=\frac{1}{\frac{1}{a^{2}}+\frac{1}{b^{2}}}=\frac{a^{2} b^{2}}{b^{2}+a^{2}}$
$\Rightarrow \frac{1}{\mathrm{P}^{2}}=\frac{\mathrm{a}^{2}+\mathrm{b}^{2}}{\mathrm{a}^{2} \mathrm{~b}^{2}}=\frac{1}{\mathrm{a}^{2}}+\frac{1}{\mathrm{~b}^{2}}$
2 is correct.
3 : Line is $\mathrm{y}=\mathrm{mx}+\mathrm{c} \Rightarrow \mathrm{mx}-\mathrm{y}+\mathrm{c}=0$
$P=\frac{|\mathrm{m}(0)-0+\mathrm{c}|}{\sqrt{\mathrm{m}^{2}+1}}$
$\Rightarrow \mathrm{P}^{2}=\frac{\mathrm{c}^{2}}{\mathrm{~m}^{2}+1} \Rightarrow \frac{1}{\mathrm{P}^{2}}=\frac{\mathrm{m}^{2}+1}{\mathrm{c}^{2}}$
3 is wrong. Only 1 and 2 are correct.
$\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ to line $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$ is $\frac{\mid \mathrm{ax}_{1}+\mathrm{by}_{1}+\mathrm{c}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}$
Here, $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)=(0,0)$ and distance $=\mathrm{P}$.
$P=\frac{|a(0)+b(0)+(c)|}{\sqrt{a^{2}+b^{2}}}$
$\Rightarrow \mathrm{P}^{2}=\frac{\mathrm{c}^{2}}{\mathrm{a}^{2}+\mathrm{b}^{2}}$
1 is correct.
2: Line is $\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1 \Rightarrow\left(\frac{1}{\mathrm{a}}\right) \mathrm{x}+\left(\frac{1}{\mathrm{~b}}\right) \mathrm{y}+(-1)=0$
$P=\frac{\left|\frac{1}{a}(0)+\frac{1}{b}(0)-1\right|}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}}=\frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}}$
$=P^{2}=\frac{1}{\frac{1}{a^{2}}+\frac{1}{b^{2}}}=\frac{a^{2} b^{2}}{b^{2}+a^{2}}$
$\Rightarrow \frac{1}{\mathrm{P}^{2}}=\frac{\mathrm{a}^{2}+\mathrm{b}^{2}}{\mathrm{a}^{2} \mathrm{~b}^{2}}=\frac{1}{\mathrm{a}^{2}}+\frac{1}{\mathrm{~b}^{2}}$
2 is correct.
3 : Line is $\mathrm{y}=\mathrm{mx}+\mathrm{c} \Rightarrow \mathrm{mx}-\mathrm{y}+\mathrm{c}=0$
$P=\frac{|\mathrm{m}(0)-0+\mathrm{c}|}{\sqrt{\mathrm{m}^{2}+1}}$
$\Rightarrow \mathrm{P}^{2}=\frac{\mathrm{c}^{2}}{\mathrm{~m}^{2}+1} \Rightarrow \frac{1}{\mathrm{P}^{2}}=\frac{\mathrm{m}^{2}+1}{\mathrm{c}^{2}}$
3 is wrong. Only 1 and 2 are correct.
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