Prove the Given Statement....3

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The discussion centers on proving the mathematical statement that for the circle defined by the equation x² + 2Ax + y² + 2By = C, the relationship a•b + c•d = -2C holds true, where a and b are x-intercepts and c and d are y-intercepts. The proof involves substituting values and utilizing the product of the roots formula, leading to the conclusion that 2ab = -2C. The participants confirm the correctness of the proof, emphasizing the straightforward nature of the problem after completing prior related problems.

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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

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Once you work the first two problems in this series, this one should be a walk in the park. :D
 
I am stuck with the second problem in the series.
 
RTCNTC said:
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
 
Nicely done.
 
If you use the product of the roots formula, you get:

$$ab+cd=-C-C=-2C$$ :D
 
Thank you, Mark. Thank you everyone.
 

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