+0  
 
0
1
2639
2
avatar+4 

find normal line equation of the curve y=sqrt(25-x^2) at the point (3,4)

crysl  Mar 23, 2015

Best Answer 

 #2
avatar+93352 
+10

$$\\y=\sqrt{25-x^2}\\\\
y=(25-x^2)^{0.5}\\\\
\frac{dy}{dx}=0.5(25-x^2)^{-0.5}\times -2x\\\\
\frac{dy}{dx}=-x(25-x^2)^{-0.5}\\\\
when \;\;x=3\\\\
\frac{dy}{dx}=-3(25-3^2)^{-0.5}\\\\
\frac{dy}{dx}=-3(16)^{-0.5}\\\\
\frac{dy}{dx}=-3\times\frac{1}{4}\\\\
\frac{dy}{dx}=-\frac{3}{4}\\\\$$

 

$$\\$Gradient of the tangent is $ -\frac{3}{4}\\\\
$Gradient of the normal is $ +\frac{4}{3}\\\\\\
\frac{4}{3}=\frac{y-4}{x-3}\\\\
4(x-3)=3(y-4)\\\\
4x-12=3y-12\\\\
$the equation of the normal at (3,4) is $
4x-3y=0$$

Melody  Mar 23, 2015
 #1
avatar+88898 
+10

y = (25 - x^2)^1/2    taking the derivative, we have

y' = (1/2)(25 -x^2)^(-1/2) (-2x) =  -x / √[25 - x^2]

And the slope of the line when x = 3 = -3/ √[25 - 3^2]  = -3/ √16 = -3/4

And the slope of the normal line will be the negative reciprocal of this = 4/3

So we have

y - 4 = (4/3)(x -3)

y = (4/3)x - 4 + 4

y = (4/3)x       and that's the equation of the normal line to the function at (3, 4)

See the pic here of the function, the tangent line to the function at (3, 4) and the normal line to the function going through the same point

https://www.desmos.com/calculator/ff3alev1sg

 

  

CPhill  Mar 23, 2015
 #2
avatar+93352 
+10
Best Answer

$$\\y=\sqrt{25-x^2}\\\\
y=(25-x^2)^{0.5}\\\\
\frac{dy}{dx}=0.5(25-x^2)^{-0.5}\times -2x\\\\
\frac{dy}{dx}=-x(25-x^2)^{-0.5}\\\\
when \;\;x=3\\\\
\frac{dy}{dx}=-3(25-3^2)^{-0.5}\\\\
\frac{dy}{dx}=-3(16)^{-0.5}\\\\
\frac{dy}{dx}=-3\times\frac{1}{4}\\\\
\frac{dy}{dx}=-\frac{3}{4}\\\\$$

 

$$\\$Gradient of the tangent is $ -\frac{3}{4}\\\\
$Gradient of the normal is $ +\frac{4}{3}\\\\\\
\frac{4}{3}=\frac{y-4}{x-3}\\\\
4(x-3)=3(y-4)\\\\
4x-12=3y-12\\\\
$the equation of the normal at (3,4) is $
4x-3y=0$$

Melody  Mar 23, 2015

35 Online Users

avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.