Question

Can someone give me the steps to find the point on the line y=2x+1 that is closest to the point (5,2)?

Answer 1

are they parallel or perpendicular?

Answer 2

they need to be orthogonal

Answer 3

perp?

Answer 4

???????

Answer 5

Yes 90deg

Answer 6

If we draw a line from (5,2) to the line y = 2x+1 in such a way that the two lines are orthogonal, the intersection of the lines is the closest point that we want. So, we need to come up with the equation for the new line.

We need two pieces of information for the equation of a line: a point and the slope. We want the lines to be orthogonal, so the slope ought to be the negative reciprocal; that is, since the slope of the first line is 2, the slope we want to use is -1/2. Luckily we already have a point to use: (5,2). Using the point-slope formula, we have
y-2 = -1/2 * (x-5)
y = -x/2 +5/2+2
y= -x/2+9/2.

Now that we have this line, the point we want is the intersection of the two lines, so set the equations equal to each other and solve for x:
-x/2 + 9/2 = 2x+1
9/2 = 5x/2 +1
7/2 = 5x/2
7/5 = x.
Plug this into one of the equations to find y:
y = 2(7/5)+1 = 14/5+1 = 19/5.
So our point should be (7/5, 19/5). You might want to double check my arithmetic, but the method should get the right answer.