# line point distance

• May 7th 2010, 08:13 PM
vvc531
line point distance
Use dot product methods to find the distance from (5,2) to the line 2x-7y=0.
• May 7th 2010, 08:58 PM
Prove It
Quote:

Originally Posted by vvc531
Use dot product methods to find the distance from (5,2) to the line 2x-7y=0.

I don't know about dot product methods, but note that the line can be rewritten as $\displaystyle y = \frac{2}{7}x$.

That means you're wanting to find the minimum distance between $\displaystyle (5, 2)$ and $\displaystyle \left(x, \frac{2}{7}x\right)$.

The distance between these points is:

$\displaystyle D = \sqrt{\left(x - 5\right)^2 + \left(\frac{2}{7}x - 2\right)^2}$

$\displaystyle = \sqrt{x^2 - 10x + 25 + \frac{4}{49}x^2 - \frac{8}{7}x + 4}$

$\displaystyle = \sqrt{\frac{53}{49}x^2 - \frac{78}{7}x + 29}$

$\displaystyle = \left(\frac{53}{49}x^2 - \frac{78}{7}x + 29\right)^{\frac{1}{2}}$.

You should now be able to find the minimum distance, by taking the derivative, setting equal to 0 and solving, then using the second derivative test.