Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    2
    ReputationRep:
    Find the Cartesian equation of the plane  \Pi that the matrix  \mathbf{M}
     \begin{bmatrix} 2 & 1 & 0 \\ 0& 1& 0 \\ 4& 2& 0 \end{bmatrix}
    maps all points to under the transformation.
    Find the Cartesian equations of the two possible spheres of radii 3 that the plane  \Pi is tangent to at the origin.
    [The equations should be in exact form not involving trigonometric functions.]

    The last bit is aarrrd, hard to visualise.
    • Study Helper
    Online

    13
    Study Helper
    (Original post by Ano123)
    hard to visualise.
    Essentially you have a plane that goes through the origin.

    Then, there are two spheres - one either side of the plane - that touch the plane at the origin. You could imagine one sitting on the plane at that point, and the other in the mirror image position - treating the plane as a mirror.
    • Community Assistant
    • Welcome Squad
    Offline

    20
    ReputationRep:
    Community Assistant
    Welcome Squad
    (Original post by Ano123)
    Find the Cartesian equation of the plane  \Pi that the matrix  \mathbf{M}
     \begin{bmatrix} 2 & 1 & 0 \\ 0& 1& 0 \\ 4& 2& 0 \end{bmatrix}
    maps all points to under the transformation.
    Find the Cartesian equations of the two possible spheres of radii 3 that the plane  \Pi is tangent to at the origin.
    [The equations should be in exact form not involving trigonometric functions.]

    The last bit is aarrrd, hard to visualise.
    Okay so for the first part I got the plane -2x+y=0, is that right? Been a while since FP4 now I'm trying to remember it.

    The last bit is not too bad as far as visualisation goes once you have the plane equation, therefore straight forward to do.
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by RDKGames)
    Okay so for the first part I got the plane -2x+y=0, is that right? Been a while since FP4 now I'm trying to remember it.

    The last bit is not too bad as far as visualisation goes once you have the plane equation, therefore straight forward to do.
    Equation of the plane is  z=2x .
    What about the next part?
    • Community Assistant
    • Welcome Squad
    Offline

    20
    ReputationRep:
    Community Assistant
    Welcome Squad
    (Original post by Ano123)
    Equation of the plane is  z=2x .
    What about the next part?
    Ah yeah, slipped up in my working and hadn't noticed I used y instead of z.

    For the next part, you imagine that the two spheres are tangent on either side of the plane. They both have the same radius of 3 and are tangent at the origin.

    So from here you can just construct a line, which is normal to plane and goes through the origin, with scalar \lambda. You can take the magnitude of this line to be 3 and solve for the two values of lambda. Sub each in and get 2 different points, these are the centres of the spheres, then proceed to put them into the sphere equation format of (x-a)^2+(y-b)^2+(z-c)^2=3^2 where the centre is (a,b,c)

    That's my thinking anyway. I like the question.
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Would you like to hibernate through the winter months?
    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

    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
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • 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

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