Question

1. $(x + 5)^{2}+(y + 7)^{2}=36$ translated 5 left, 4 down

Explanation:

Step1: Recall the standard form of a circle's equation

The standard form of a circle's equation is \((x - h)^2+(y - k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius. For the given equation \((x + 5)^2+(y + 7)^2=36\), we can rewrite it as \((x-(- 5))^2+(y-(-7))^2 = 6^2\). So the center is \((-5,-7)\) and the radius \(r = 6\).

Step2: Apply the translation rules

When a point \((x,y)\) is translated \(a\) units left and \(b\) units down, the new point \((x',y')\) is given by \(x'=x - a\) and \(y'=y - b\). Here \(a = 5\) and \(b = 4\). For the center \((h,k)=(-5,-7)\), the new \(h'\) is \(-5-5=-10\) and the new \(k'\) is \(-7 - 4=-11\).

Step3: Write the new equation of the circle

Using the standard form \((x - h')^2+(y - k')^2=r^2\), substituting \(h'=-10\), \(k'=-11\) and \(r = 6\), we get \((x+10)^2+(y + 11)^2=36\)

Answer:

\((x + 10)^2+(y+11)^2=36\)