Question

In the diagram below,AB is the diameter of the circle with centre P.Point C lies on the y-axis.Given the coordinates of A and B are (-6,3) and (2,-3) respectively ,calculate

a)the coordinates of P





b)radius of the circle





c)the coordinates of C





d)the area of the circle in term of

`\pi`
​

Answer 1

The coordinates of P are (-2,0)

The radius of the circle is 5.

Explanation:

Analytic geometry

The diagram shows a circle with center P, and two points A(-6,3) and B(2,-3) that form the diameter of the circle.

a)

The center of the circle lies at the midpoint of A and B. The midpoint (xm,ym) can be calculated by:

`\displaystyle x_m=(x_1+x_2)/(2)`

`\displaystyle y_m=(y_1+y_2)/(2)`

Substituting x1=-6, x2=2, y1=3, y2=-3:

`\displaystyle x_m=(-6+2)/(2)=(-4)/(2)=-2`

`\displaystyle y_m=(3-3)/(2)=0`

Thus, the coordinates of P are (-2,0)

b) The radius of the circle is the distance from the center to any point in its circumference. We can use the distance from P to A or B indistinctly.

Given two points A(x1,y1) and P(x2,y2), the distance between them is:

`d=√((x_2-x_1)^2+(y_2-y_1)^2)`

Substituting x1=-6, x2=-2, y1=3, y2=0:

`r=√((-2+6)^2+(0-3)^2)`

`r=√(4^2+(-3)^2)`

`r=√(16+9)=√(25)=5`

The radius of the circle is 5.