Find the distance from point `P=(2,-1)` to the line `3x+4y=0.`
- print Print
- list Cite
Expert Answers
calendarEducator since 2012
write609 answers
starTop subjects are Math, Science, and History
Let `P=(x_0,y_0)` be point and `l` line with equation `Ax+By+C=0` , then the distance between point `P` and line `l` is
`d(P,l)=(|Ax_0+By_0+C|)/(sqrt(A^2+B^2))`
In your case `x_0=2,` `y_0=-1,` `A=3,` `B=4` and `C=0.` Hence we have
`d=(|3cdot2+4cdot(-1)|)/(sqrt(3^2+4^2))=(|6-8|)/(sqrt(9+16))=|-2|/sqrt25=2/5`
The distance is `2/5.`
Related Questions
- Find if the distance between the point (2, y) and the line 3x-4y+5 = 0 is 12 units. Find y
- 2 Educator Answers
- find the distance from the point A(-2,1,2) to the plane 3x-2y+5z+1 =0
- 1 Educator Answer
- Find the locus of a moving point equidistant from the line 2x+y=10 and 3x+4y=6.
- 1 Educator Answer
- Find the distance between the point (2,-5) and the line 3x-5y+13= 0
- 2 Educator Answers
- Solve y''+4y'+3y=0, y(0)=2, y'(0)=-1
- 1 Educator Answer
Another way of doing this is to find the point of intersection Q of the perpendicular drawn from the point P(2, -1) to the line 3x + 4y = 0 and then determine the distance between the two points P and Q.
Write 3x + 4y = 0 in slope intercept form
4y = -3x
y = (-3/4)x
The slope of a line perpendicular to this is 4/3. The equation of a a line with slope 4/3 and passing through (2, -1) is
4/3 = (y + 1)/(x - 2)
4x - 8 = 3y + 3
4x = 3y + 11
The point of intersection of 4x = 3y + 11 and 3x + 4y = 0 can be determined by solving the two equations.
Substitute x = (-4/3)y in 4x = 3y + 11
4*(-4/3)y = 3y + 11
-16y = 9y + 33
25y = -33
y = -33/25
x = 44/25
The distance between (2, -1) and (44/25, -33/25) is
D = `sqrt((2 - 44/25)^2 + (-1 + 33/25)^2)`
= `(1/25)sqrt((50 - 44)^2 + (-25 + 33)^2)`
= `(1/25)sqrt(6^2 + 8^2)`
= `10/25`
= 0.4
The distance between the point (-2, 1) and the line 3x + 4y = 0 is 0.4
Student Answers