# Calculus problem involving finding coordinates

1. Aug 13, 2013

### 5ymmetrica1

1. The problem statement, all variables and given/known data

Using calculus, find the coordinates of the point on the line y =-2x+5, which is closest to the origin, and the corresponding value of D

2. Relevant equations

y = -2x +5

3. The attempt at a solution

I know I need to find a line that is perpendicular to the line of the equation, but I'm not sure how to find the equation of this line so that I can make the equations equal to each other.

Something like

d(x) = x2+y2
= x2 + (-2x+5)2
= x2 + 4x2+25
= 5x2 + 25

am I on the right track?

2. Aug 13, 2013

### besulzbach

Assuming that $$(-2x + 5)^{2} = 4x^{2} + 25$$ is terrible.
I'm not good at calculus but this problem is quite easy, even though a mistake like this one can ruin your answer.
Be careful that
$$(-2x + 5)^{2} = (-2x + 5)(-2x + 5) =\ ...$$

3. Aug 13, 2013

### LCKurtz

To add to besulzbach's reply, note that what you are calculating is the distance squared from the origin to a point on the line. (Not that there's anything wrong with that.)

4. Aug 13, 2013

### mark.watson

If the line y1 = -2x + 5 has a slope of -2, then a line perpendicular will have a slope of 1/2.

Since you are trying to find the points closest to the origin, the equation of the perpendicular line is y2 = (1/2)x

This should be enough...

5. Aug 13, 2013

### LCKurtz

...although whether it qualifies as a method "using calculus" is uncertain.

6. Aug 13, 2013

### mark.watson

True... he mentioned wanting to find a line perpendicular, so I went with that.

Last edited: Aug 13, 2013
7. Aug 13, 2013

### 5ymmetrica1

Ah that's embarrassing! I forgot to expand the problem, I been away from math for to long haha. thanks for the replies guys!

So
is it 4x2 - 10x -10x + 25
= 4x2-20x+25

if f(x)= 4x2-20x+25
then f'(x) = 8x -20
and f''(x) = 8

if 8x - 20 = 0
8x= 20
x= 20/8
x=2.5

y = -2x + 5
y = -2(2.5) + 5
y = 0

So coordinates are (2.5, 0)
How does this look

8. Aug 13, 2013

### mark.watson

Using Calculus:

f(x) = x2 + y2, where y = -2x + 5

so, f(x) = 5x2 - 20x + 25;
therefore, f'(x) = 10x - 20.

If f'(x) = 0 when f(x) is minimum, then

0 = 10x - 20, so x = 2.

Then, y = y(x) = y(2) = -2(2) + 5 = 1.

The coordinate is (2, 1).

NOTE: This agrees with the other "non-Calculus" method I used earlier.

Last edited: Aug 13, 2013
9. Aug 13, 2013

### LCKurtz

You left out an $x^2$ from the other term.

10. Aug 14, 2013

### besulzbach

$$f(x) = 5x^{2} - 20x + 25$$
Is NOT the function for the distance, it is the formula to find the square of the distance!
Use that $f(x)$ to make a function $d(x)$ that gives you the distance for any input value of $x$.
$$d(x) = \sqrt{5x^{2} - 20x + 25}$$
Any point $(x, (-2x + 5))$ will be $\sqrt{5x^{2} - 20x + 25}$ from the origin. Now you can calculate D as you wanted.

11. Aug 23, 2013

### 5ymmetrica1

So the next question asks me to make a conjecture and prove it with algebra

So I know that I need to do the same thing as with the first part but using

y = ax + b

f(x)=x2+y2

f(x)=x2+(ax+b)(ax+b)

f(x)=x2+a2 x2+2axb+b2

then do I differentiate this?

I'm not sure how to make a conjecture based on this though