Question

Which of the following equations are equations of circles? Choose all that apply

Answer 1

Answer:

Step-by-step explanation:

The work form of a circle is

$(x-h)^2+(y-k)^2=r^2$

The standard form of a circle is

$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$

This standard form is a circle if A and C are the same number, there is a plus sign in between the x-squared and the y-squared terms, and the F value, when moved to the other side of the equals sign, is not negative.  That F value represents the radius which is a measure of distance, and distances cannot be negative.

Looking at the first choice in your list, on the right side of the equals sign we have a negative 4.  That means that that equation is not a circle.

Looking at the second choice in your list, we have a 2x^2 and a y^2, where A = 2 and C = 0.  This is not a circle, either, since A and C have to be the same number in a circle, and they are not.

Looking at the third choice in your list, we have x^2 and y^2 with the same A and C value of 1, and if we move the F value of -5 over, we have a radius of 5, and there is a + sign in between the x^2 and the y^2 terms.  Choice 3 is a circle.

Looking at the fourth choice, we have that A = 3 and C = 3, there is a + sign in between the x^2 and the y^2 terms, and if we move the -6 over to the other side, the radius is a positive number.  Choice 4 is also a circle.

Looking at the last choice, there is a - sign in between the x^2 and the y^2 terms.  Even though A and C are both the same number, 1, the - sign makes that possibly a hyperbola, but definitely NOT a circle.