Determine whether the statement If displaystyle{D}ne{0}{quadtext{or}quad}{E}ne{0}, then the graph of displaystyle{y}^{2}-{x}^{2}+{D}{x}+{E}{y}={0} is a hyperbolais true or false. If it is false, explain why or give an example that shows it is false.

zi2lalZ 2021-02-15 Answered
Determine whether the statement If D0orE0,
then the graph of y2x2+Dx+Ey=0 is a hyperbolais true or false. If it is false, explain why or give an example that shows it is false.
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Expert Answer

estenutC
Answered 2021-02-16 Author has 81 answers

Step 1
We have given a statement:
If D0orE0,
then graph of y2x2+Dx+Ey=0 is a hyperbolais.
Step 2
We know the general form of conic section:
Ax2+Bxy+Cy2+Dx+Ey+F=0
To find the type of conic section we solve for B24AC:
(i) If B24AC<0 then the conic section is ellipse.
(ii) If B24AC<0andA=C,B=0 then we have a perfect circle.
(iii) If B24AC=0, then we have a parabola.
B24AC>0, then we have a hyperbola
Hyperbola defined as:
(xh)2a2+(yk)2b2=1
Where (h, k) are center.
When E,D=0,then(h,k)=(0,0)
Then the center will be at (0,0)
Step 3
Hence, the given statement is incorrect since the given condition is not mandatory for a conic section to be a hyperbola.

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