Write the standard form of the equation of the circle that is tangent to the line x=3 and has center at (-2,-7).
Answer: (x+2)^2+(y+7)^2=49
Is this correct?
did you make a quick sketch?
should it not be obvious from your sketch, that the radius would be 5 ?
so the equation should have been:
(x+2)^2 + (y+7)^2 = 25
Yes, that is correct!
To explain how to get the standard form of the equation of the circle, let's break it down step by step.
Step 1: Find the radius of the circle.
Since the circle is tangent to the line x=3, the distance between the line and the center of the circle is equal to the radius. In this case, the center of the circle is (-2,-7) and the distance between x=3 and (-2,-7) is 5. Therefore, the radius of the circle is 5.
Step 2: Determine the equation of the circle.
The standard form of the equation of a circle with center (h,k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2.
In this case, the center of the circle is (-2,-7) and the radius is 5. Plugging these values into the equation, we get:
(x - (-2))^2 + (y - (-7))^2 = 5^2
(x + 2)^2 + (y + 7)^2 = 49
So, the standard form of the equation of the circle is (x + 2)^2 + (y + 7)^2 = 49, which is the correct answer.