Which one of the following differential equations represents the family of straight lines which are at unit distance from the origin a) `(y-x(dy)/(dx))^2=1-((dy)/(dx))^2` b) `(y+x(dy)/(dx))^2=1+((dy)/(dx))^2` c)`(y-x(dy)/(dx))^2=1+((dy)/(dx))^2` d) `(y+x(dy)/(dx))^2=1-((dy)/(dx))^2`

Options

A. `(y-x(dy)/(dx))^(2)=1-((dy)/(dx))^(2)`
B. `(y+x(dy)/(dx))^(2)=1+((dy)/(dx))^(2)`
C. `(y+x(dy)/(dx))^(2)=1+((dy)/(dx))^(2)`
D. `(y+x(dy)/(dx))^(2)=1-((dy)/(dx))^(2)`

Answer

C

Explanation

y= mx +c ( Equation of straight line)
` (dy)/(dx) =m and mx -y+c=0` is at unit distance from origin.
` (|m (0) -(0)+c|)/(sqrt(m^(2) +(-1)^(2)) =1 Rightarrow c = sqrt(1 +m^(2))`
Now :
` [ y -x (dy)/(dx)]^(2) = [mx +c - xm]^(2) Rightarrow c^(2) =1 +m^(2)`
also,
`[y +x (dy)/(dx)]^(2)= [ mx +c+mx]^(2) = [2mx + sqrt(1 +m^(2))] ^(2)`
also, ` 1 - ((dy)/(dx))^(2) =1 - m^(2) and 1 + ((dy)/(dx))^(2) =1 + m^(2)`
` Rightarrow [ y -x (dy)/(dx)]^(2) =1 + ((dy)/(dx))^(2)`