precalculus.complete Table.

Question

1.  
   • Column 2: Indicate the values for A, B, C, D, E, and F by comparing the equation in Column 1 to the general equation for a conic section: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.
   • Column 3: Evaluate B2 – 4AC and indicate if this value is greater than, less than, or equal to zero. 
   • Column 4: Use the value of B2 – 4AC and the style of the equation to categorize each of the equations as representing a circle, parabola, ellipse, or hyperbola.
   • Column 5: Find a factored form of the expression on the left-hand side of the equation. If the expression is not factorable, write cannot be factored.
   •  

 

Answer 1

Compare the Equation with the general Equation of conic : Ax² + Bxy + Cy² + Dx + Ey + F = 0

Conditions :

B² – 4AC < 0 then Equation represents Circle or Ellipse

B² – 4AC > 0 then Equation represents Hyperbola

B² – 4AC = 0 then Equation represents Parabola

Second Degree Equation	Values for A, B, C, D, E, F		Evaluate B² – 4AC. Indicate if >, <, or = 0.	Type of Conic Section	Factored Form (if possible)
x²+ y² = 4	A = 1 B = 0 C = 1	D = 0 E = 0 F = –4	– 4 < 0	Circle	cannot be factored
x² – y²= 1	A = 1 B = 0 C = -1	D = 0 E = 0 F = –1	4 > 0 & A+C = 0	Rectangular Hyperbola	cannot be factored
x² + y = 4	A = 1 B = 0 C = 0	D = 0 E = 1 F = –4	0-4*1*0 = 0	Parabola
x²– 9y² = 1	A = 1 B = 0 C = -9	D = 0 E = 0 F = –1	0-4*1*-9 = 36 > 0	Hyperbola
	A = 1/9 B = 0 C = 1	D = 0 E = 0 F = –1	0-4*(1/9)*1 < 0	Ellipse
y – x² = 1	A = -1 B = 0 C = 0	D = 0 E = 1 F = –1	0-4*(-1)*1 > 0	Hyperbola	y= x² + 1
9x² – 4y² = 36	A = 9 B = 0 C = -4	D = 0 E = 0 F = -36	0 - 4*9*(-4) = 144 > 0	Hyperbola