My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
October 29th, 2012, 08:14 AM   #1
Member
 
Joined: Oct 2012

Posts: 94
Thanks: 0

normal to the curve/normal to the circle

The line that is normal to the curve y = x^2 + 2x - 3 at (1,0) intersects the curve at what other point?


Find the points on the curve x^2 + xy + y^2 = 7 (a) where the tangent is parallel to the x-axis and (b) where the tangent is parallel to the y axis.

For both of them, I differentiated. For the first, I found the slope and figured out the line. Don't know where to go from there, though. If someone could really walk me through this it would be appreciated.
cheyb93 is offline  
 
October 29th, 2012, 08:16 AM   #2
Member
 
Joined: Oct 2012

Posts: 94
Thanks: 0

Re: normal to the curve/normal to the circle

Oops, I wrote the wrong problem for the second one. The second problem should be: Find the two points where the curve x^2 + xy + y^2 = 7 crosses the x axis and show that the tangents to the curve at these points are parallel. What is the common slope of these tangents?
cheyb93 is offline  
October 29th, 2012, 11:45 AM   #3
Math Team
 
Joined: Sep 2007

Posts: 2,409
Thanks: 6

Re: normal to the curve/normal to the circle

Quote:
Originally Posted by cheyb93
The line that is normal to the curve y = x^2 + 2x - 3 at (1,0) intersects the curve at what other point?
You say you have found the equation of the normal line, in the form y= mx+ b, I presume. To determine whete that line itersects the curve, set them equal, or [/latex]x^2+ (2- m)+ b- 3= 0[/tex]. You already know one solution so it should be easy to find the other..


Quote:
Find the points on the curve x^2 + xy + y^2 = 7 (a) where the tangent is parallel to the x-axis and (b) where the tangent is parallel to the y axis.

For both of them, I differentiated. For the first, I found the slope and figured out the line. Don't know where to go from there, though. If someone could really walk me through this it would be appreciated.
Any line parallel to the x- axis is of the form "y= constant", which has slope 0. Any line parallel to the x- axis is of the form "x= constant", which does not have a "slope".
HallsofIvy is offline  
October 29th, 2012, 12:14 PM   #4
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 520

Math Focus: Calculus/ODEs
Re: normal to the curve/normal to the circle

1.) We are given:



Implicitly differentiate with respect to :







This is the slope of the normal line, now us the point-slope formula to get the equation of the line:





Now, substitute for in the equation of the parabola:









Discard the known root and we have:

and so

Hence, the normal line also intersects the parabola at .

2.) We are given:



To find where this curve crosses the x-axis, we set , and find:





Thus, the two x-intercepts are:



Implicitly differentiating the given curve with respect to x, we find:









Hence, we have shown that the common slope of the two tangents is -2.
MarkFL is offline  
October 29th, 2012, 12:59 PM   #5
Member
 
Joined: Oct 2012

Posts: 94
Thanks: 0

Re: normal to the curve/normal to the circle

I just have one question...I guess more about the algebra.

When I try to compute x-1 = -4x^2 - 8x +12, I get 0 = -4x^2 - 9x + 13

I'm not exactly sure why, in your answer, all of the signs are opposite.
cheyb93 is offline  
October 29th, 2012, 01:05 PM   #6
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 520

Math Focus: Calculus/ODEs
Re: normal to the curve/normal to the circle

The two are equivalent, just multiply through by -1, or equivalently, move everything to the other side.
MarkFL is offline  
October 29th, 2012, 01:10 PM   #7
Member
 
Joined: Oct 2012

Posts: 94
Thanks: 0

Re: normal to the curve/normal to the circle

Quote:
Originally Posted by MarkFL
The two are equivalent, just multiply through by -1, or equivalently, move everything to the other side.

Is that why dx/dy (2x + 2) = 1 becomes -dx/dy = -1/2(x+1) as well? Thanks. Was a little confused about the negatives in that one too.
cheyb93 is offline  
October 29th, 2012, 01:11 PM   #8
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 520

Math Focus: Calculus/ODEs
Re: normal to the curve/normal to the circle

Yes, exactly. We needed to solve for -dx/dy to get the slope of the normal line.
MarkFL is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
circle, curve or normal, normal



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Standard normal density curve Exfactor Algebra 4 January 2nd, 2014 04:20 PM
Equation of normal to a curve with parameter 940108 Calculus 5 May 21st, 2013 03:50 AM
normal vector, legnth of curve in mathematica kapital Math Software 0 August 4th, 2012 05:07 AM
Finding the equation of the normal to the curve help please BellEnd Calculus 2 January 12th, 2011 09:31 AM
Find the eq of tangent and normal to the curve varunnayudu Trigonometry 1 January 4th, 2010 12:17 AM





Copyright © 2019 My Math Forum. All rights reserved.