If a certain line intersects the origin and is perpendicular to the line with the equation `y = 2x+5` at point P, what is the distance from the origin to point P ?

Options

A. `2.24`
B. `2.45`
C. `2.67`
D. `3.25`

Answer

A

Explanation

If the two lines are perpendicular, then their slopes are negative reciprocals.
The slope of `y=2x+5` is 2, so the slope of the other line is `-(1)/(2)`. That line goes through the origin, so its y-intercept is 0, and therefore its equation is `y=-(1)/(2)x`. Point P - the intersection of these lines - is the point that satisfies both equations :
`y=2x+5` and `y=-(1)/(2)x`
`2x+5=-(1)/(2)x`
`4x+10=-x`
`5x=-10`
`x=-2`
`y=-(1)/(2)x=-(1)/(2)(-2)=1`
So point P is (-2, 1), and the distance from the origin to point P is
`OP=sqrt((-2)^(2)+1^(2))=sqrt(5)~~2.24`