Determine whether the statement, 'I noticed that depending on the values for A and C, assuming that they are not both zero, the graph of Ax^{2} + Cy^{2} + Dx + Ey + F = 0 can represent any of the conic sections', makes sense or does not make sense, and explain your reasoning.

Question
Conic sections
asked 2020-10-21
Determine whether the statement, 'I noticed that depending on the values for A and C, assuming that they are not both zero, the graph of \(\displaystyle{A}{x}^{{{2}}}+{C}{y}^{{{2}}}+{D}{x}+{E}{y}+{F}={0}\) can represent any of the conic sections', makes sense or does not make sense, and explain your reasoning.

Answers (1)

2020-10-22
Given, assuming that both A and C are non-zero, then the graph of \(\displaystyle{A}{x}^{{{2}}}+{C}{y}^{{{2}}}+{D}{x}+{E}{y}+{F}={0}\) can represents any of the conic sections. We have to check whether the statement is true of not. Since the general form of the conic is \(\displaystyle{A}{x}^{{{2}}}+{C}{y}^{{{2}}}+{D}{x}+{E}{y}+{F}={0}\) If both A and C are zero then the equation of the section becomes linear which is a contradiction. The equation of a conic section is quadratic. Hence if the values of A and B are non-zero then the graph of \(\displaystyle{A}{x}^{{{2}}}+{C}{y}^{{{2}}}+{D}{x}+{E}{y}+{F}={0}\) can represent any of the conic sections. Hence the statement is true.
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