Prove the Given Statement....3

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Discussion Overview

The discussion revolves around proving a mathematical statement related to the properties of a circle defined by the equation x² + 2Ax + y² + 2By = C, specifically focusing on the relationship between the x-intercepts (a, b) and y-intercepts (c, d) of the circle. The scope includes mathematical reasoning and problem-solving strategies.

Discussion Character

  • Mathematical reasoning
  • Homework-related
  • Technical explanation

Main Points Raised

  • One participant presents the problem statement and requests proof of the relationship a•b + c•d = -2C.
  • Another participant suggests that solving previous problems in the series will make this problem easier.
  • A participant expresses difficulty with an earlier problem in the series, indicating a potential challenge in understanding the current problem.
  • A participant provides a derivation that leads to the conclusion 2ab = ab + cd = -2C, using substitutions and manipulations of the original equation.
  • Another participant confirms the derivation using the product of the roots formula, arriving at the same conclusion of ab + cd = -2C.
  • A participant expresses gratitude for the assistance received in the discussion.

Areas of Agreement / Disagreement

Participants appear to agree on the derivation leading to the conclusion, but there is no explicit consensus on the overall approach or any unresolved aspects of the proof process.

Contextual Notes

Some participants reference earlier problems in the series, suggesting that understanding those may be crucial for tackling the current problem. There may be assumptions about the familiarity with the product of the roots formula that are not explicitly stated.

mathdad
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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

How is this started?
 
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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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