When finding distance which is better 'General Form' or 'Substitution'; given point and line?

Question

I am given a line y = 2/3x - 5 and a point (1, 4); I need to find the distance.

The first example from Saxon Calculus, would solve this like so:
f-1(x) = f(x) to find x1:
[3/2x + 5/2 = 2/3x - 5] So x = -9;

f(x1) to find y1: 
[y = 2/3(-9) - 5] So y = -11; 

Now using the distance formula for (x1, y1), (1, 4): 
d = √[(-9 - 1)^2 + (-11 - 4)^2]  So the distance d = √325 = 25√13
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The second example would suggest otherwise; it solves this as follows:
First, I know the following:
1. m = -(A/B)
2. C = Bb or b = C/B

-> Ax + By = C
-> 2x - 3y = 15 or [Ax + -By = (-Bb)]
-> 2x - 3y - 15 = 0
-> |Ax + By + C| / √(A^2 + B^2)
-> |2(1) + (-3)(4) - 15| / √[2^2 + (-3)^2] = 25/√13

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25√13 ≠ 25/√13
did I missing something, what did I or the book do wrong?
~~~~~~~~~~~~~~~~~~~~~~~~~~~

I checked the first example and point (-9, -11) satisfies both slope-intercept equations...

Sqdancefan · Accepted Answer

Don't know what Saxon Calculus tells you, but I do know the f-1(x) line equation you have neither goes through the point (1,4) nor intersects the given line at right angles. The line you need to be using is
.. f'(x) = -3/2x + 11/2
This has a slope that is the negative inverse of the slope of the given line, and it has (1, 4) as a solution.

Using this, you will find the point of intersection with the given line to be (x, y) = (63/13, -23/13) and the distance from this point to the given point as
.. d = ||(1, 4) - (63/13, -23/13)|| = ||(-50/13, 75/13)|| = 25/13||(-2, 3)|| = 25/13*√((-2)^2 + 3^2)
.. d = (25/13)√13 = 25/√13

stockett · Answer

there are a number of techniques on the thank you to discover the equation of a line. I choose first of all looking the equation utilising element-slope, (on account which you're given 2 factors), and then convert to typical type. First, discover the slope of the line utilising the two factors: (y2 -y1)/(x2 - x1) (5 - 2)/(2 - -one million) 3/3 = one million= m Now, plug in the slope for element-slope type: y - y1 = m(x - x1) y - 2 = m(x - -one million) y = one million(x +one million) +2 y = x + one million + 2 y = x + 3 Now convert to typical type: flow the x + 3 to the left-hand portion of the equation. y = x + 3 -1x + y - 3 = 0

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When finding distance which is better 'General Form' or 'Substitution'; given point and line?

2 Answers