Question

Find the distance between the point (-3, -4) and line 2y = -3x + 6.

a. square root 13
b. 23
c. 14
d. 23 square root 13 over 13

Answer 1

Option D.

Explanation:

The distance between a point and a line is

`Distance=(|Ax_1+By_1+C|)/(√(A^2+B^2))`

The given equation of line is

`2y=-3x+6`

It can be rewritten as

`3x+2y-6=0`

Using the above formula the distance between (-3,-4) and is

`Distance=(|3(-3)+2(-4)-6|)/(√(3^2+2^2))`

`Distance=(|-9-8-6|)/(√(9+4))`

`Distance=(|-23|)/(√(13))`

`Distance=(23)/(√(13))`

Rationalize the expression.

`Distance=(23)/(√(13))* (√(13))/(√(13))`

`Distance=(23√(13))/(√(13))`

Therefore, the correct option is D.

Answer 2

a. square root 13

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