Question

23
the equation of a circle is shown.
$(x - 3)^2 + (y - 2)^2 = 4$
the circle is translated 2 units to the right and 4 units up and then is dilated by a factor of 3.
what is the equation of the new circle?

Explanation:

Step1: Identify original circle properties

The standard form of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius. For the original circle \((x - 3)^2 + (y - 2)^2 = 4\), the center is \((3,2)\) and the radius \(r=\sqrt{4} = 2\).

Step2: Apply translation

Translating 2 units right (add 2 to \(x\)-coordinate of center) and 4 units up (add 4 to \(y\)-coordinate of center). New center: \(h'=3 + 2=5\), \(k'=2 + 4 = 6\).

Step3: Apply dilation

Dilating by a factor of 3. The radius scales by the dilation factor: new radius \(r'=2\times3 = 6\), so \(r'^2=6^2 = 36\).

Step4: Write new circle equation

Using the standard form with new center \((5,6)\) and new radius squared 36: \((x - 5)^2 + (y - 6)^2 = 36\).

Answer:

\((x - 5)^2 + (y - 6)^2 = 36\)