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#1 2016-11-11 14:40:15

Monox D. I-Fly
Registered: 2015-12-02
Posts: 857

[ASK] About Circles in Coordinates

3. A(a,b), B(-a,-b), and C is plane XOY. P moves along with curve C. If the multiplication product of PA's and PB's gradients are always k, C is a circle only if k = ...?
4. The radius of a circle which meets X-axis at (6,0) and meets the curve

at one point is ....
5. A circle meets the line x + y = 3 at (2,1). It also meets the point (6,3). Its radius is ....

I have no idea how to do number 3, or even the meaning.

For number 4, I substituted x = 6 and y = 0 to the equation

and got

For number 5, I substituted both coordinates to the circle equation and got 2a + b = 10.

Please help how to continue each number.


#2 2016-11-11 16:18:19

From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [ASK] About Circles in Coordinates


What is a and b for 5?

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.


#3 2016-11-12 04:31:16

bob bundy
Registered: 2010-06-20
Posts: 8,137

Re: [ASK] About Circles in Coordinates

hi Monox D. I-Fly

Q3.  P is (x,y), a point on an unknown curve C in the x-y plane.  Form an expression for each gradient in terms of x, y, a, and b and multiply to make k.

For a circle the coefficients of x^2 and y^2 must be equal.  That's enough to fix k.

Q4.  What do you mean by 'meets'?  I assumed this meant "has the x axis as a tangent at that point" but that doesn't give a sensible diagram.  So do you mean 'goes through' the point?

Also is that y = (root3) times x or root(3 times x)  ?

If the former then this is a straight line at 60 degrees so the angle bisector of this line and the x axis will be at 30 degrees and the centre lies on this bisector.

If the latter, it seems you need to differentiate as the circle and curve will have a common tangent at the point.  That makes the question very complicated (well it is for me smile ).  I've tried a diagram and cannot get such a tangent at all.  Best I can get with my software is centre at about (2.78, -1.34) buit don't rely on that as accurate.

Q5.  Again the word meets.  Just touches or goes through ?

Later edit:  Q5.  If we assume that x+y+3 is a tangent to the circle at (2,1), then we can calculate the equation through (2,1) perpendicular to x+y=3.

The centre will lie on this line so you can get a second equation connecting a and b.

bobbym:   The circle has centre (a,b) in conic section notation.


Last edited by bob bundy (2016-11-12 06:06:59)

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