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    A circle C, with centre (a,b) and radius 5, touches the x axis at (4,0).

    a) I calculate a to be 4 and b to be 5, hence, centre (4,5)
    b) To form a cartesian equation i used (x-4)^2 + (y-5)^2 = 25, multiplied out and tided up as standard.
    x^2 + y^2 - 8x - 10y + 16 = 0

    c) This is where i have my query:
    A tangent to the circle, drawn from the point P(8,17), touches the circle at T. Find, to 3 significant figures, the length of PT.
    My query comes from the fact that i have no idea as to the coordinates of T, or the gradient of the line. How am i supposed to find either of those? Surely there are many tangents to the curve, any idea how i am supposed to calculate which passes through P?
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    Let T be the point (x,y)
    now make up a straight-line eqn using P(8,17) as the other point on the line.
    The coords (x,y) of T also satisfy the eqn of your circle.
    You now have two eqns in two unknowns - albeit one of them is a quadratic - use substitution.
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    Better method:

    CTP is a right-angled triangle, so use Pythagoras to find PT (CT is clearly 5 and you can find PC using Pythagoras again)
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    (Original post by mathsexams)
    Better method:

    CTP is a right-angled triangle, so use Pythagoras to find PT (CT is clearly 5 and you can find PC using Pythagoras again)
    Yeah - that's faster
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    (Original post by Fermat)
    Let T be the point (x,y)
    now make up a straight-line eqn using P(8,17) as the other point on the line.
    The coords (x,y) of T also satisfy the eqn of your circle.
    You now have two eqns in two unknowns - albeit one of them is a quadratic - use substitution.
    It's not working. If i let T have coordinates (x,y), the equation of the line simply cancels itself out. If i let T have coordinates (a,b), i have equations:
    Line:
    bx - 17x - 8b = ay - 8y - 17a
    Circle, with a,b substituted:
    a^2 + b^2 - 8a - 10b + 16 = 0

    Giving me 4 variables.

    Could I ask how you know CTP is a right angled triangle?
    Anyone have a full working as i'm getting more and more confused.
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    (Original post by Gaz031)
    It's not working. If i let T have coordinates (x,y), the equation of the line simply cancels itself out. If i let T have coordinates (a,b), i have equations:
    Line:
    bx - 17x - 8b = ay - 8y - 17a
    Circle, with a,b substituted:
    a^2 + b^2 - 8a - 10b + 16 = 0

    Giving me 4 variables.
    I was too quick - sorry.
    mathsexams method is much better.
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    There's no need to be sorry, any attempt at help is appreciated.
    I'm just irritated by how i can walk the other questions, including 2 for 7 marks, and the only one i can't do is for a measily 3.
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    (Original post by Gaz031)
    ...

    Could I ask how you know CTP is a right angled triangle?
    Anyone have a full working as i'm getting more and more confused.
    CPT is a right-angled triangle because PT is a tangent to the circle, and CT is a radius of the circle and radii and tangents are always at right angles.
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    Here's some working.
    Attached Images
     
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    Thanks for the working but:
    PT^2 = PC^2 - PT^2 ?
    Did you mean PC^2 - CT^2 ?
    If so, where did you obtain CT^2 from?

    Thanks.
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    (Original post by Gaz031)
    Thanks for the working but:
    PT^2 = PC^2 - PT^2 ?
    Did you mean PC^2 - CT^2 ?
    If so, where did you obtain CT^2 from?

    Thanks.
    Oh drat! I think I'll go to bed I'm obviously not awake
    Yes that should be PC^2 - CT^2
    and CT = 5 because CT is a radius and the radius of the circle we know is 5.
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    Ahh, yes, it's certainly 5, so CT^2 is 25.
    Sorry for being such an idiot, obscure circle and vector questions are my weakpoint in P3.
    Thanks to all who gave help.
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    What other conditions were given for the first part of the question? Because radius 5 and centre (4, -5) also works.

    I'm just curious as I'm working through it myself for practice and I haven't done many circle questions.
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    is there a p3 exam coming up or summat?
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    (Original post by lollipop)
    is there a p3 exam coming up or summat?
    In November, perhaps?

    Ben
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    well thats a bit patronising!

    How am I spose to know???

    L
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    (Original post by lollipop)
    well thats a bit patronising!

    How am I spose to know???

    L
    I'm sorry? I wasn't being patronising!

    Ben
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    (Original post by lollipop)
    is there a p3 exam coming up or summat?
    Not as far as i know. I'm doing A2 maths and AS+A2 FM among other subjects this year so i want to get modules done and enforce my knowledge by past exam papers.
    Plus i'm aware that P3 knowledge will be a perequisite P4,P5,P6. I wouldn't want to be like someone who sat AS FM last year, he had to take P4 but hasn't even done P3 yet now! Although i'll be sitting a lot of maths this year at least i can do it in some sort of order.
    I could be sitting D1/P2 in November, though my teacher isn't quite sure whether it's allowed. If not, i'll be sitting D1/Hopefully P4 at christmas.

    As for the other information in the question, i'll type it out. It's a fairly low mark question though:

    The circle C, with centre (a,b) and radius 5, touches the x axis at (4,0),
    a) Write down the value of a and the value of b. (1)
    There's a diagram that shows nothing extra apart from emphasizing the fact that the circle only touches the x axis once, and does not go through it.
    This leads to the conclusion the distance from the centre to the x axis must be 5, it cannot be greater than 5 as this is the radius, it cannot be less than 5 or else it would go through the axis twice at other points.
    Hence, the radius is a straight vertical line at the point it touches (4,0). The radius is 5, therefore the y coordinate b is 5.
    It touches the x axis at 4, and the radius would be a vertical line with no chance in x. Therefore the x coordinate a is 4.
    a = 4, b = 5,

    b) (x-4)^2 + (y-5)^2 = 25, is an equation, or you can multiply out and simplify.
    c) Explained well above by earlier posters.
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    Is it clear from the diagram that b is +5 or is it just an arbitary line and circle demonstrating what "touches the x axis" means?
 
 
 
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