Answer:
The equation of the circle is
Step-by-step explanation:
The complete question is
If the coordinates of the endpoints of a diameter of the circle are known, the equation of a circle can be found. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius, which is the distance from the center to either endpoint of the diameter. Finally use the center-radius form to find the equation.
Find the center-radius form of the circle having the points (1,4) and (-7,6) as the endpoints of a diameter.
Consider that, if both points are the endpoints of a diameter, the center of the circle is the point that is exactly in the middle of the two points (that is, the point whose distance to each point is equal). Given points (a,b) and (c,d), by using the distance formula, you can check that the middle point is the average of the coordinates. Hence, the center of the circle is given by
.
We will find the radius. Recall that the radius of the circle is the distance from one point of the circle to the center. Recall that the distance between points (a,b) and (c,d) is given by . So, let us use (1,4) to calculate the radius.
.
REcall that given a point . The equation of a circle centered at the point
is
In our case, and
. Then, the equation is
