Respuesta :
Answer:
a. square root 13
Step-by-step explanation:
We need to find the equation of the perpendicular line to 2y= -3x+6 that passes through the given point (-3, -4) and then find the intersect of the two lines and finally calculate the distance beteen this intersect and the point (-3,-4)
Be careful the algebra sadly is a bit messy but here we go:
2y = -3x + 6
y= -3/2 + 3
The slope of the perpendicular line is the negative inverse of (-3/2) which is 2/3
The perpendicular equation passing through (-3,-4) has the equation:
y-(-4)/(x-(-3)) = 2/3
y +4 = 2/3*x +2
y = 2/3 * x - 2
Now have a system of 2 equations with two unknowns
which are
y= -3/2 *x + 3
y= 2/3 *x -2
solving -3/2 *x + 3 = 2/3 * x -2
x = 30/13
annd then substituting in any of the equations to get y
y= -18/39 = -6/13
then the distance is given by the square root of the difference of the x values squared + the difference of y values squared ( Pythagoream Theorem):
sqrt (( (30/13 -(-3))2 + ( -6/13 +4 )^2 ))
sqrt ( 9^2/13^2 + 46^2/13^2)
1/13 * square root of (81 + 46^2) = 1/13 sqrt(2197) (2197 =13 to the cube)
therefore we have
= ((1/13) * 13) * sqrt 13 = sqrt 13
Answer:
Option D.
Step-by-step explanation:
The distance between a point and a line is
[tex]Distance=\dfrac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}[/tex]
The given equation of line is
[tex]2y=-3x+6[/tex]
It can be rewritten as
[tex]3x+2y-6=0[/tex]
Using the above formula the distance between (-3,-4) and is
[tex]Distance=\dfrac{|3(-3)+2(-4)-6|}{\sqrt{3^2+2^2}}[/tex]
[tex]Distance=\dfrac{|-9-8-6|}{\sqrt{9+4}}[/tex]
[tex]Distance=\dfrac{|-23|}{\sqrt{13}}[/tex]
[tex]Distance=\dfrac{23}{\sqrt{13}}[/tex]
Rationalize the expression.
[tex]Distance=\dfrac{23}{\sqrt{13}}\times \dfrac{\sqrt{13}}{\sqrt{13}}[/tex]
[tex]Distance=\dfrac{23\sqrt{13}}{\sqrt{13}}[/tex]
Therefore, the correct option is D.
