MHB Prove the Given Statement....3

[mathdad]

Suppose that the circle x^2 + 2Ax + y^2 + 2By = C has two intercepts, a and b, and two y-intercepts, c and d.
Prove that a•b + c•d = -2C

How is this started?

 

[MarkFL]

Once you work the first two problems in this series, this one should be a walk in the park. :D

 

[mathdad]

I am stuck with the second problem in the series.

 

[Greg]

Suppose that the circle x^2 + 2Ax + y^2 + 2By = C has two intercepts, a and b, and two y-intercepts, c and d.
Prove that a•b + c•d = -2C

$$A=-\frac{a+b}{2}$$

and the point (a, 0) is on the circle. Plug and chug:

$$-ab=C$$

Get sneaky:

$$2ab=-2C$$

...and using the result from part 2:

$$2ab=ab+cd=-2C$$

QED

 

[mathdad]

Nicely done.

 

[MarkFL]

If you use the product of the roots formula, you get:

$$ab+cd=-C-C=-2C$$ :D

 

[mathdad]

Thank you, Mark. Thank you everyone.