To find the diameter of the circle (Two dimensional Geometry)

1. Apr 6, 2010

snshusat161

1. The problem statement, all variables and given/known data
A line meets the coordinates axes in A and B. A circle is circumscribed about the triangle OAB. If m and n are the distances of the tangent to the circle at the origin from the point A and B respectively, the diameter of the circle is:

(a) m(m + n)

(b) m + n

(c) n(m + n)

(d) (1/2)(m + n)

2. Relevant equations

nothing so special. I've included some equations with the attempt I've submitted here.

3. The attempt at a solution

let coordinates of A and B are (a, 0) and (0, b) respectively. Since 0A and AB are perpendicular to each other therefore the center of the circle lies at the midpoint of AB and AB is the diameter.

so the length of diameter is $$\sqrt{}a^2 + b^2$$

2. Apr 6, 2010

lanedance

i agree with you if the m = OA & n = OB, however the way i read it is (which may be wrong as the question is a little ambiguous):
- the edge of the circle passes through the origin
- draw the tangent line to the circle through the origin
- now the distance m will be the perpindicular from the tangent line to A and similar for n

that makes it a fair bit trickier, but you may be able to make some headway by looking at similar trinagles (and it gives me one of the multiple choice answers above)

Last edited: Apr 6, 2010
3. Apr 6, 2010

snshusat161

Thanks, You have a given a nice hint and it may really solve the problem.

Last edited: Apr 6, 2010
4. Apr 6, 2010

lanedance

from reading the red bit below
though as i said, I don't think its written very clear, but thats how i'd interpret it

5. Apr 6, 2010

snshusat161

I still can't understand this sentence but the hint you have given is working so this sentence should mean the same.

6. Apr 6, 2010

lanedance

it may help to know if you have a line & a point, the shortest line joining them will be perpidicular to the original line

7. Apr 6, 2010

snshusat161

Can you explain this a little more. Actually I'm in need of such tricks which can solve problems in very less time. We only get 1 minute to solve each problem. And all questions are objective type question so I don't need the actual method but the final answer.

8. Apr 6, 2010

lanedance

at first glance you could rule out a) & c) as the units are incorrect

then the key is always to draw a good diagram, for this case as follows
- coordinate axes
- line through A & B (make these different lengths to help)
- draw the hypotenuse which is the diameter of the circle, as you mentioned
- mark the centre of the circle
- try & draw a cricle as best as possible around the triangle
- now draw the tangent through the origin
- then draw the radius from the centre of circle to the origin
- note its is perpindicular to the tangent line as it must be
- now draw lines perpinduclar to the tanget through A & B to give m & n (they are parallel to the radius)

so you should see one of m or n is larger than the radius & one smaller, which should lead you to your question

sounds like a lot, but you shoudl be able to do it pretty quickly

9. Apr 6, 2010

snshusat161

thanks. That dimensional analysis was very good trick to rule out some options but rest I'm unable to do it on exam because here they don't permit us to use anything except our pen. No rounder, no divider, no scale, no calculator, nothing. And to draw and find out exactly is very uncertain

10. Apr 6, 2010

lanedance

if you draw well enough it should be obvious, if C is the centre of the circle then AC = r = CB, which with a little thought, show:
r = (m + n)/2

but just looking a) & c) fall away dimensionly, or if you assume m,n>1 they will be much bigger than your diagram, so invalid

similarly
b) = 2 d)
so hopefully you can rule one out easily

Last edited: Apr 6, 2010
11. Apr 6, 2010

snshusat161

I've drawn for this question. Let the point of intersection of tangent from O and tangent from A be P and point of intersection of tangent from O and tangent from B be Q then ABQP looks like a rectangle. So it give a misconception that OC (radius) is equal to m and n. So If someone follow the diagram then he will surely mark on the option (d).

12. Apr 6, 2010

snshusat161

oh sorry! Question asked for diameter. You are right.

13. Apr 6, 2010

Cilabitaon

Just for future reference; any point along the circumference of a circle that creates a tangent to the circle will:
a)be a distance $$r$$(radius) from the origin;
b)form a right-angled triangle along the diameter of the circle with any other point on the circumference.

14. Apr 6, 2010

snshusat161

Thanks, It will help me a lot on my examination.