Question

which of the following is not a true statement? choose the incorrect statement. a. the equation, (x - 5)^2 + y^2 = 5, represents a circle in standard form. b. the equation, x^2 + 4y^2 - 8x + 12y + 5 = 0, represents a circle in general form. c. a circle of the form x^2 + y^2 = 4 has a center at (0,0). d. you can graph the equation of a circle if you know the radius and the center of the circle.

Explanation:

Step1: Recall circle - form equations

The standard - form of a circle is \((x - a)^2+(y - b)^2=r^2\), where \((a,b)\) is the center and \(r\) is the radius. The general form of a circle is \(x^{2}+y^{2}+Dx + Ey+F = 0\).

Step2: Analyze option A

The equation \((x - 5)^2+y^{2}=5\) is in the standard form \((x - a)^2+(y - b)^2=r^2\) with \(a = 5\), \(b = 0\), and \(r=\sqrt{5}\), so it represents a circle in standard form.

Step3: Analyze option B

The equation \(x^{2}+4y^{2}-8x + 12y+5 = 0\) is not a circle. The general form of a circle has the coefficients of \(x^{2}\) and \(y^{2}\) equal. Here, the coefficient of \(x^{2}\) is 1 and the coefficient of \(y^{2}\) is 4, so it represents an ellipse, not a circle.

Step4: Analyze option C

The equation \(x^{2}+y^{2}=4\) is in the standard form \((x - 0)^2+(y - 0)^2 = 2^2\), so the center is \((0,0)\) and the radius \(r = 2\).

Step5: Analyze option D

If we know the center \((a,b)\) and radius \(r\) of a circle, we can graph it using the standard - form \((x - a)^2+(y - b)^2=r^2\). We can plot the center point \((a,b)\) and then draw a circle with radius \(r\) around it.

Answer:

B. The equation, \(x^{2}+4y^{2}-8x + 12y + 5=0\), represents a circle in general form.