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**Monox D. I-Fly****Member**- Registered: 2015-12-02
- Posts: 933

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

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.

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

Hi;

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.**

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,168

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 ). 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.

Bob

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

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