The equations representing the circle having a diameter of 12 units and a center lying on y-axis are <u>x² + (y – 3)² = 36</u>, and <u>x² + (y + 8)² = 36</u>, making <u>options 1, and 5</u> the two choices.
In the question, we are looking for the equations of the circle having a diameter of 12 units and a center lying on y-axis.
The standard form of the equation of a circle is (x - h)² + (y - k)² + r², where (h, k) is the center of the circle, and r is its radius.
A circle having a diameter of 12 units, will have a radius of 12/2 = 6 units. Thus, r = 6.
A circle having the center on the y-axis will have a center as (0, k) as all points on the y-axis have x-coordinate 0.
Thus, the equation of the circle, with a diameter of 12 units and the center lying on the y-axis takes the form:
x² + (y - k)² = 6², or, x² + (y - k)² = 36.
From the given options, the equations <u>x² + (y – 3)² = 36</u>, and <u>x² + (y + 8)² = 36</u>, are of this form.
Thus, the equations representing the circle having a diameter of 12 units and a center lying on y-axis are <u>x² + (y – 3)² = 36</u>, and <u>x² + (y + 8)² = 36</u>, making <u>options 1, and 5</u> the two choices.
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The provided question is incomplete. The complete question is:
"Which equations represent circles that have a diameter of 12 units and a center that lies on the y-axis? Select two options.
- x² + (y – 3)² = 36
- x² + (y – 5)² = 6
- (x – 4)² + y² = 36
- (x + 6)² + y² = 144
- x² + (y + 8)² = 36."
is the nth number, 
is the first number, n is the number of terms and d is the difference of the two consecutive numbers.