# Normals with no point

• Jun 16th 2010, 09:46 AM
sammy28
Normals with no point
hi all,

Im not too sure how to answer this question:

Find the equations of the normals to the curve xy=3 which are parallel to the line 3x - y - 2 = 0

obviously the equations need to be

$y=3x \pm c$

but how do you solve for c?

presumably you need to find the gradient function ie
$y'=-\frac{3}{x^2}$

do you set

$3x-2=y'$

thanks
sammy
• Jun 16th 2010, 10:20 AM
earboth
Quote:

Originally Posted by sammy28
hi all,

Im not too sure how to answer this question:

Find the equations of the normals to the curve xy=3 which are parallel to the line 3x - y - 2 = 0

obviously the equations need to be

$y=3x \pm c$

but how do you solve for c?

presumably you need to find the gradient function ie
$y'=-\frac{3}{x^2}$

do you set

$3x-2=y'$

thanks
sammy

1. The given line has the equation $y = 3x-2$

2. If the normal to the curve has the slope m = 3 then the curve must have the gradient $m = -\frac13$

3. That means $y' = -\frac3{x^2} = -\frac13~\implies~|x|=3$

4. The normals with m = 3 have to pass through the points P(-3, -1) and Q(3, 1).

Use the point-slope-formula of the straight line to get their equations.

5. The given line: green. The 2 normals in red:
• Jun 16th 2010, 10:56 AM
sammy28
thanks earboth (Clapping)

why is it so obvious when your shown the steps (Hi).

Using $y-y{_1}=m(x-x{_1})$, i see that $c=\pm 8$

cheers
sammy