Answer:
The equation of a circle can be represented in the form (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.
A diameter of a circle is a straight line passing through the center of the circle and has endpoints on the circle.
Given that the equation of the diameter of a circle is x+y-14=0 and 2x-y-4=0, we can find the center of the circle by solving for the intersection of these two lines.
We can find the intersection point of these lines by solving the system of equations using the substitution method:
x+y-14=0 and 2x-y-4=0
x = 14/3, y = 10/3
So the center of the circle is (14/3, 10/3)
Since we know that the point (3,4) lies on the circle, we can use the distance formula to find the radius of the circle.
r = sqrt((x-h)^2 + (y-k)^2)
r = sqrt((3-(14/3))^2 + (4-(10/3))^2) = sqrt(5)
Therefore, the equation of the circle is (x-14/3)^2 + (y-10/3)^2 = 5^2
or (x-14/3)^2 + (y-10/3)^2 = 25
So, the equation of the circle passing through point (3,4) and having equation of its diameter x+y-14=0 and 2x-y-4=0 is (x-14/3)^2 + (y-10/3)^2 = 25