Question

Identify the ellipses, represented by equations, whose eccentricities are less than 0.5.

Answer 1

Ellipses that are represented by equations, whose eccentricities are less than 0.5 are:

• (A) 49x2 -98x + 64yz +256y -2,831 = 0
• (B) 81x2 -648x +100yz +200y -6,704 = 0    
• (F) 64x2 +512x +81y2 -324y -3,836 = 0

What is an equation?

• An equation is a formula in mathematics that expresses the equality of two expressions by connecting them with the equals sign =.
• The word equation and its cognates in various languages may have somewhat different definitions; for example, in French, an équation is defined as including one or more variables, whereas in English, an equation is any well-formed formula consisting of two expressions linked by an equals sign.

To identify the ellipses:

For the standard-form equation of an ellipse: Ax^2+Bx+Cy^2+Dy+E=0

We can define:

• p=min(A,C)
• q=max(A,C)

Then the eccentricity can be shown to be:

• e=(1-p/q)
• p/q=1-e^2

For Eccentricity<0.5, we want:

• p/q>3/4

Checking the values of p/q for the given equations, we have p/q= equations (A), (B), and (F) are of ellipses with eccentricity < 0.5.

Therefore, ellipses that are represented by equations, whose eccentricities are less than 0.5 are:

• (A) 49x2 -98x + 64yz +256y -2,831 = 0
• (B) 81x2 -648x +100yz +200y -6,704 = 0    
• (F) 64x2 +512x +81y2 -324y -3,836 = 0

Know more about equations here:

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The complete question is given below:

Identify the ellipses, represented by equations, whose eccentricities are less than 0.5.

(A) 49x2 -98x + 64yz +256y -2,831=0            

(B) 81x2 -648x +100yz +200y -6,704 =0                                      

(C) 6x2 -12x +54y2 +108y -426 =0                  

(D) 49x2 + 196x +36y2 +216y -1,244 =0                                        

(E) 4x +32x +25y2 - 250y +589 =0                

(F) 64x2 +512x +81y2 -324y -3,836 =0