Turn on thread page Beta

Curves Defined Implicitly Help!!!!! watch

Announcements
    • Thread Starter
    Offline

    0
    ReputationRep:
    Hi,

    Im stuck on these 2 questions on the OCR board P3:

    1) x^2 - 2xy + 2y^2 = 5 find the tangent to the curve at the point (1,2)

    2) find the points on the curve 4x^2 + 2xy - 3y^2 = 39 at which the tangent is parallel to one of the axes.

    Cheers

    Streety
    Offline

    10
    ReputationRep:
    1) 2x - 2[y+x(dy/dx)] + 4y(dy/dx) = 0
    Now substitute x=1 and y=2 to find dy/dx, i.e. the gradient of the tangent.

    2) 8x + 2[y+x(dy/dx)] - 6y(dy/dx) = 0
    The tangent is parallel to an axis when it touches the maximum point. So, substitute dy/dx=0 to get the equation.
    • Thread Starter
    Offline

    0
    ReputationRep:
    Cheers helped loads
    Offline

    10
    ReputationRep:
    Glad I could help.
    Offline

    0
    ReputationRep:
    (Original post by streetyfatb)
    Hi,

    Im stuck on these 2 questions on the OCR board P3:

    1) x^2 - 2xy + 2y^2 = 5 find the tangent to the curve at the point (1,2)

    2) find the points on the curve 4x^2 + 2xy - 3y^2 = 39 at which the tangent is parallel to one of the axes.

    Cheers

    Streety
    This is probably wrong, but no one else's answered and I'm bored.

    1)
    x^2 - 2xy + 2y^2 = 5 differentiate
    2x-2y -2x*(dy/dx)+4y(dy/dx)=0 substitute (1,2)
    2-4-2(dy/dx)+8(dy/dx)=0
    6dy/dx=2
    dy/dx=1/3
    Tangent: y=mx+c so m=1/3
    y=x/3+c substitute (1,2)
    2=1/3+c
    c=5/3
    y=x/3+5/3
    3y-x=5 is the tangent.

    2)

    4x^2 + 2xy - 3y^2 = 39
    8x+2y+2x(dy/dx)-6y(dy/dx)=0
    dy/dx(2x-6y)=-2y-8x
    dy/dx=(8x+2y)(6y-2x)
    Tangent parallel to axis when gradient is zero or infinite. It is therefore parallel to axis when: y=-4x (i.e. dy/dx is 0, parallel to x axis), when x,y=0 (gradient infinite, parallel to y axis) and when y=3x, again gradient infininte.
    • Thread Starter
    Offline

    0
    ReputationRep:
    Cheers guy, still not with qu 2 yet so if u could explain 2 me it would be much appricated.


    Cheers

    Streety
    Offline

    10
    ReputationRep:
    The gradient is defined as tan(m), where m is the angle the tangent makes with the x-axis.

    For the line to be parallel to the x-axis, it must touch the maximum point of the curve, i.e. when m=180. That is: dy/dx = tan(m) = tan180 = 0.

    For the line to be parallel to the y-axis, it must make a right angle with the x-axis. That is: dy/dx = tan(m) = tan90 = undefined/infinity.

    I don't think you're expected to use the second case though, since the question asked for an equation of the tangent parallel to one of the axes.
    Offline

    0
    ReputationRep:
    (Original post by streetyfatb)
    Cheers guy, still not with qu 2 yet so if u could explain 2 me it would be much appricated.


    Cheers

    Streety
    You start off with: 4x^2 + 2xy - 3y^2 = 39
    Then differentiate everything with respect to x: 8x+2y+2x(dy/dx)-6y(dy/dx)=0
    You know factorise the dy/dx parts: 8x+2y+(2x-6y)=0
    Then shift the other terms to the other side: (2x-6y)(dy/dx)=-2y-8x
    Then divide through by (2x-6y) to get dy/dx by itself: dy/dx=(-8x-2y)/(2x-6y)
    Then get rid of the minus signs: dy/dx=(8x+2y)/(6y-2x)
    You've now obtained an expression for dy/dx.
    For dy/dx to be parallel to the axis, there are two cases: parallel to the x axis, and parallel to the y axis. Parallel to the x axis means that no matter what the shift in x is, the shift in y is zero. Hence dy/dx=0. For this to happen, (8x+2y)/(6y-2x) has to be zero. i.e. y=-4x.
    If the tangent is parallel to the y axis, that means for no x increase at all, you get as large a y increase as you like. This means dy/dx is infinite. So (8x+2y)/(6y-2x) is inifite. That means 6y-2x has to be zero; that can happen in two ways: either the'yre both zero, or x=3y.
    So in total, you've got three set of solution: one infinite set of solutions where y=-4x, one infininte set where x=3y, and one isolated case, where x=y=0
    Offline

    13
    ReputationRep:
    And a bit more... The expression for the gradient is zero when y=-4x and infinite when x=3y but it's important not to forget that not all points that satisfy these also satisfy the original equation.

    4x^2 + 2xy - 3y^2 = 39

    Straight away x=y=0 does not satisfy this so this cannot be one of the points.

    Subbing in x=3y for infinite gradient:
    4(3y)^2 + 2(3y)y - 3y^2 = 39
    36y^2 + 6y^2 - 3y^2 = 39
    39y^2 = 39
    y = +-1 => x = +-3

    Now y=-4x for zero gradient:
    4x^2 + 2x(-4x) - 3(-4x)^2 = 39
    4x^2 - 8x^2 - 48x^2 = 39
    -52x^2 = 39
    This will lead to an imaginary solution so dy/dx =/= 0 so the tangent is never parallel to the x axis.

    So the points are (3,1) and (-3,-1) where we have infinite gradient and the tangent is parallel to the y axis.
 
 
 
Turn on thread page Beta
Updated: September 9, 2004

University open days

  • University of the Arts London
    MA Performance Design and Practice Open Day Postgraduate
    Thu, 24 Jan '19
  • Brunel University London
    Undergraduate Experience Days Undergraduate
    Sat, 26 Jan '19
  • SOAS University of London
    Postgraduate Open Day Postgraduate
    Sat, 26 Jan '19
Poll
Are you chained to your phone?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Equations

Best calculators for A level Maths

Tips on which model to get

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.