Question

Consider the following statements :
1. The length p of the perpendicular from the origin to the line `ax+by=c` satisfies the relation `p^(2)=c^(2)/(a^(2)+b^(2))`
The length p of the pependicular from the origin to the line `x/a+y/b=1` satisfies the relation `1/p^(2)=1/a^(2)+1/b^(2)`.
3. The length p of the perpendicular from the origin to the line `y=mx+c` satisfies the relation `1/p^(2)=(1+m^(2)+c^(2))/c^(2)`
Which of the above is/are correct ?
A. 1, 2 and 3
B. 1 only
C. 1 and 2 only
D. 2 only

Answer 1

Correct Answer - C
1 : We know, the perpendicular distance from
`(x_(1), y_(1))` to line `ax+by+c=0` is `(|ax_(1)+by_(1)+c|)/sqrt(a^(2)+b^(2))`
Here, `(x_(1), y_(1))=(0, 0)` and distance `=P`.
`:. P=(|a(0)+b(0)+(c)|)/sqrt(a^(2)+b^(2))`
`implies P^(2)=c^(2)/(a^(2)+b^(2))`
`:.` 1 is correct.
2 : Line is `x/a+y/b=1 implies (1/a)x+(1/b)y+(-1)=0`
`P=(|1/a (0)+1/b(0)-1|)/sqrt(1/a^(2)+1/b^(2))=1/sqrt(1/a^(2)+1/b^(2))`
`=P^(2)=1/(1/a^(2)+1/b^(2))=(a^(2)b^(2))/(b^(2)+a^(2))`
`implies 1/P^(2)=(a^(2)+b^(2))/(a^(2)b^(2))=1/a^(2)+1/b^(2)`
2 is correct.
3 : Line is `y=mx+c implies mx-y+c=0`.
`:. P=(|m(0)-0+c|)/sqrt(m^(2)+1)`
`implies P^(2)=c^(2)/(m^(2)+1) implies 1/P^(2)=(m^(2)+1)/c^(2)`
3 is wrong.
`:.` Only 1 and 2 are correct.