Question

If (P=(-3,5)) and (Q=(1,9)), find the equation of the circle that has segment PQ as a diameter.

((x - [?])^2 + (y - [?])^2 = [?])

a) (x - 2)^2 + (y - 7)^2 = 25)
b) (x + 2)^2 + (y + 7)^2 = 25)
c) (x - 3)^2 + (y - 5)^2 = 25)
d) (x + 3)^2 + (y + 5)^2 = 25)

Answer 1

To find the equation of a circle that has segment PQ as a diameter, we can use the midpoint formula to find the coordinates of the midpoint of PQ, which will be the center of the circle. Using the coordinates of P=(-3,5) and Q=(1,9), the equation of the circle is (x - 3)^2 + (y - 5)^2 = 25.

Step-by-step explanation:

To find the equation of the circle that has segment PQ as a diameter, we can use the midpoint formula to find the coordinates of the midpoint of PQ, which will be the center of the circle. The midpoint formula is given by:

Midpoint = ( (x1 + x2)/2, (y1 + y2)/2 )

Using the coordinates of P=(-3,5) and Q=(1,9), we can calculate the midpoint as follows:

Midpoint = ( (-3 + 1)/2, (5 + 9)/2 ) = ( -1/2, 7 )

Therefore, the equation of the circle that has segment PQ as a diameter is:

(x + 1/2)^2 + (y - 7)^2 = ( (1 - (-1/2))^2 + (9 - 7)^2 )/4 = 25/4

Simplifying this equation, we get:

(x + 1/2)^2 + (y - 7)^2 = 25/4

So, the correct answer is option c) (x - 3)^2 + (y - 5)^2 = 25).