Calculus problem involving finding coordinates

[5ymmetrica1]

Homework Statement

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

Homework Equations

y = -2x +5

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?

 

[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
$$(-2x + 5)^{2} = (-2x + 5)(-2x + 5) =\ ...$$

 

[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.)

 

[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...

 

[LCKurtz]

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...

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

 

[mark.watson]

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

 

[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

 

[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.

 

[LCKurtz]

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

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

 

[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!

(...) and the corresponding value of D

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.

 

[5ymmetrica1]

thanks for your help guys
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