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P3 Past paper question query. watch

1. 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?
2. 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.
3. 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)
4. (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
5. (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.
6. (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.
7. 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.
8. (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.
9. Here's some working.
Attached Images

10. 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.
11. (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.
12. 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.
13. 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.
14. is there a p3 exam coming up or summat?
15. (Original post by lollipop)
is there a p3 exam coming up or summat?
In November, perhaps?

Ben
16. well thats a bit patronising!

How am I spose to know???

L
17. (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
18. (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.
19. 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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