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Consider a circle whose equation is x^2 + y^2 – 2x – 8 = 0. Which statements are true? Check all that apply.


A.The radius of the circle is 3 units.

B.The center of the circle lies on the x-axis.

C.The center of the circle lies on the y-axis.

D.The standard form of the equation is (x – 1)² + y² = 3.

E.The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

Verified Answer :

Answer:  The correct options are

A. The radius of the circle is 3 units.

C. The center of the circle lies on the y-axis.

E. The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

Step-by-step explanation:  The given equation of the circle is

\(x^2+y^2-2x-8=0~~~~~~~~~~~~~~~~~~~(i)\)

We are given to select the TRUE statements from the given options.

The standard form of a circle with radius r units and center (h, k) is given by

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

From equation (i), we have

\(x^2+y^2-2x-8=0\\\\\Rightarrow x^2+(y^2-2x+1)-8-1=0\\\\\Rightarrow x^2+(y-1)^2=9\\\\\Rightarrow (x-0)^2+(y-1)^2=3^2.\)

comparing the above equation with the standard equation of a circle, we get

center, (h, k) = (0, 1)  and  radius, r = 3 units.

since the x co-ordinate of the center (0, 1) is 0, so it lies on the Y-axis.

Now, we have

\(x^2+y^2=9\\\\\Rightarrow (x-0)^2+(y-0)^2=3^2.\)

So, the radius of this circle is 3 units.

That is, the radius of the given circle is same as the radius of the circle whose equation is \(x^2+y^2=9,\)

Thus, the standard equation of the given circle is

\(x^2+(y-1)^2=3^2,\) the center lies on the Y-axis, the radius is 3 units and the radius of the given circle is same as the radius of the circle whose equation is \(x^2+y^2=9.\)

Options (A),(C) and (E) are correct.

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