Shortest Distance between an Equation and a Line

[Abraham Chileshe]

1. Give an equation with an undefine slope i.e x = 4. How can i find the shortest distance between this equation and a point M(3, -9) since this point doesn't lie within this equation?

2. How can i find the coordinates of intersection between the line 4x − 12y − 5 = 0, and the equation of a line with points M(−4, 1) и N(−4, 5

 

[stapel]

(1) If it's a vertical line, in which direction would the shortest measurement be from the vertical line to the point?

(2) What is meant by the backwards N between the two points M and N?

 

[The Highlander]

1. Give an equation with an undefine slope i.e x = 4. How can i find the shortest distance between this equation and a point M(3, -9) since this point doesn't lie within this equation?

2. How can i find the coordinates of intersection between the line 4x − 12y − 5 = 0, and the equation of a line with points M(−4, 1) и N(−4, 5

1. Draw the line on a graph then plot the point. You should see immediately what the shortest distance is!

2. Sorry, don't understand your expression: "points M(−4, 1) и N(−4, 5".

 

[Dr.Peterson]

2. How can i find the coordinates of intersection between the line 4x − 12y − 5 = 0, and the equation of a line with points M(−4, 1) и N(−4, 5

First, as I understand it, "и" means "and", in Cyrillic languages. So this is asking for the intersection of the line with equation 4x − 12y − 5 = 0 and the line containing the points M(−4, 1) and N(−4, 5).

So, find the equation of that second line (using its slope and either point -- this will be very easy, though not typical), and then solve the resulting system of equations.

When you do this, please show your work and ask any specific questions it raises.

 

[Steven G]

I think that students should try looking at the points that lie on a line and see if they can come up with a rule that works for both points.

ex 1: (2,5) and (8,11). It seems that the y value is always 3 more than the x value

Is or is always means =
y value means y and x value means x.

Putting this together we get y = x+3. That is the equation of the line

ex 2: (3,6) and (-5,-10). It seems to me that the y value is double the x value. Therefore the equation of the line if y=2x.

In your example, (−4, 1) and (−4, 5), it seems that x is always -4. Therefore the equation of the line is x=-4.

As long as your rule is linear and it works for both points, then you have the equation.

 

[The Highlander]

1. Give an equation with an undefine slope i.e x = 4. How can i find the shortest distance between this equation and a point M(3, -9) since this point doesn't lie within this equation?

2. How can i find the coordinates of intersection between the line 4x − 12y − 5 = 0, and the equation of a line with points M(−4, 1) и N(−4, 5

In addition to all of the sound advice provided above (and accepting @Dr.Peterson's definition of the symbol "и"), if you follow my suggested approach for Part 1, then, on the same sketch/graph (created in Part 1), simply plot the two points (M & N) and draw a (straight) line through them both.

You should then observe a distinct similarity between your new line and the previous one (x=4) that should provide you with the x-coordinate of the intersection you need to find (as it will be the same for every point on that line).

Plugging that x-value into your 4x − 12y − 5 = 0 equation will then provide the y-coordinate of the intersection.

Or you might (for the sake of practice) wish to also draw the 4x − 12y − 5 = 0 line (y=\(\displaystyle \frac{1}{3}\)x-\(\displaystyle \frac{5}{12}\)) and just see where they cross each other?