Math Is Fun Forum

niharika_kumar

Member

From: Numeraland

Registered: 2013-02-12

Posts: 1,062

circle

If two distinct chords of a circle

drawn from a point (p,q) is bisected by x-axis ,then which of the following is correct.

a)

b)

c)

d)

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friendship is tan 90°.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: circle

hi Nikarita,

EDIT:  See next two posts.  New diagram in preparation.

Bob

Last edited by Bob (2015-03-04 21:08:11)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: circle

Hi Bob;

Should (p,q) be on the circle?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: circle

Yes it should.  Just realised that myself.

New diagram:

Here the centre is now G.  A is still (p,q).  Chords are BA and CA.  Midpoints E and D respectively.  F is the origin.

A and F lie on the circle (and B and C of course!)

Haven't got a method yet, but here's a suggestion:

Introduce letters for the coordinates of B and C.

Work out the coordinates of E and D.

Set their y coordinates to zero and see what happens.

More later.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: circle

Let A be the point (p,q). The circle is

[list=*]
[*]

[/*]
[/list]

Clearly the circle passes through the origin O and the point A and has radius OA/2. Thus OA is the diameter the circle.

Because of symmetry we can assume WLOG that p, q > 0 (i.e. A is in the first quadrant). Let a chord drawn from A intersect the x-axis at (t,0). One end of the chord is A; if the axis bisects it, the other end would be (P,Q) where:

[list=1]
[*]

[/*]
[*]

[/*]
[/list]

Susbstiting (x,y) = (P,Q) into the equation of the circle and simplifying gives the following quadratic equation in t:

[list=*]
[*]

[/*]
[/list]

This equation has two real and distinct solutions in t; therefore its discriminant is strictly positive:

[list=*]
[*]

[/*]
[/list]

Last edited by Olinguito (2015-03-04 21:40:50)

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Bassaricyon neblina

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: circle

Do I get a medal for solving this tough problem?

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Bassaricyon neblina

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: circle

hi Olinguito

I'm considering the medal but you may have to do a little more to satisfy my limited brain .

As my diagram doesn't have OA as diameter, I'm thinking yours is different to mine.  Are you able to post a diagram please?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: circle

Why doesn't your diagram have OA as diameter? The equation of the circle is

[list=*]
[*]

[/*]
[/list]

which can be rearranged as

[list=*]
[*]

[/*]
[/list]

which is a circle centred on

= midpoint of OA with radius

.

Does not that make OA the diameter?

Last edited by Olinguito (2015-03-04 22:28:48)

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Bassaricyon neblina

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: circle

Why?  Because I didn't spot that property.  Whoops!

I have a diagram.  Now I'm happy. 

Medal awarded.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: circle

Thanks Bob!

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Bassaricyon neblina

niharika_kumar

Member

From: Numeraland

Registered: 2013-02-12

Posts: 1,062

Re: circle

thanks.

and
HAPPY HOLI to all of you.

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friendship is tan 90°.

niharika_kumar

Member

From: Numeraland

Registered: 2013-02-12

Posts: 1,062

Re: circle

can you give me a diagram of the double ordinate of a parabola.
I am not able to visualize it.

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friendship is tan 90°.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: circle

hi Nikarita

I'm not sure what you mean.  Quadratic graphs are parabolas and if you draw a line like y = 5, it will usually cut the graph in two places.  Have a look at this:

http://www.mathsisfun.com/data/function … 1=x^2-5x+3

You'll have to enter the line yourself.  I cannot find the right syntax to put in the link above directly.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob