Answer:
![\huge\boxed{Center= (-3,4) , Radius = 5\sqrt{2} }](https://tex.z-dn.net/?f=%5Chuge%5Cboxed%7BCenter%3D%20%28-3%2C4%29%20%2C%20Radius%20%3D%205%5Csqrt%7B2%7D%20%7D)
Step-by-step explanation:
<u><em>Given equation is </em></u>
<u><em></em></u>![x^2 + y^2 + 6x-8y - 25 = 0](https://tex.z-dn.net/?f=x%5E2%20%2B%20y%5E2%20%2B%206x-8y%20-%2025%20%3D%200)
Adding 25 to both sides
![x^2 + y^2 +6x-8y = 25](https://tex.z-dn.net/?f=x%5E2%20%2B%20y%5E2%20%2B6x-8y%20%3D%2025)
Completing squares
![x^2 +6x +y^2 - 8y = 25\\(x)^2-2(x)(-3) + (y)^2 - 2(x)(4) = 25](https://tex.z-dn.net/?f=x%5E2%20%2B6x%20%2By%5E2%20-%208y%20%3D%2025%5C%5C%28x%29%5E2-2%28x%29%28-3%29%20%2B%20%28y%29%5E2%20-%202%28x%29%284%29%20%3D%2025)
Both of their "b" is -3 and 4 respectively
So, adding (-3)² => 9 and (4)² => 16 to both sides
![(x+3)^2 + (y-4)^2 = 25 + 9 + 16\\(x+3)^2 + (y-4)^2 = 50\\(x-(-3))^2 + (y-4)^2 = (5\sqrt{2)^2}](https://tex.z-dn.net/?f=%28x%2B3%29%5E2%20%2B%20%28y-4%29%5E2%20%3D%2025%20%2B%209%20%2B%2016%5C%5C%28x%2B3%29%5E2%20%2B%20%28y-4%29%5E2%20%3D%2050%5C%5C%28x-%28-3%29%29%5E2%20%2B%20%28y-4%29%5E2%20%3D%20%285%5Csqrt%7B2%29%5E2%7D)
Comparing it with
, where center = (h,k) and radius = r.
We get:
Center = (-3,4)
Radius = ![5\sqrt{2}](https://tex.z-dn.net/?f=5%5Csqrt%7B2%7D)
Answer:
778.688
.........................
<u>Given</u>:
The quadratic equation is ![x^{2}=-5 x-3](https://tex.z-dn.net/?f=x%5E%7B2%7D%3D-5%20x-3)
We need to determine the solutions of the quadratic equation.
<u>Solution</u>:
Let us solve the equation to determine the value of x.
Adding both sides of the equation by 5x and 3, we get;
![x^{2}+5 x+3=0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B5%20x%2B3%3D0)
The solution of the equation can be determined using quadratic formula.
Thus, we get;
![x=\frac{-5 \pm \sqrt{5^{2}-4 \cdot 1 \cdot 3}}{2 \cdot 1}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-5%20%5Cpm%20%5Csqrt%7B5%5E%7B2%7D-4%20%5Ccdot%201%20%5Ccdot%203%7D%7D%7B2%20%5Ccdot%201%7D)
![x=\frac{-5 \pm \sqrt{25-12}}{2 }](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-5%20%5Cpm%20%5Csqrt%7B25-12%7D%7D%7B2%20%7D)
![x=\frac{-5 \pm \sqrt{13}}{2 }](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-5%20%5Cpm%20%5Csqrt%7B13%7D%7D%7B2%20%7D)
Thus, the two roots of the equation are
and ![x=\frac{-5- \sqrt{13}}{2 }](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-5-%20%5Csqrt%7B13%7D%7D%7B2%20%7D)
Hence, the solutions of the equation are
and ![x=\frac{-5- \sqrt{13}}{2 }](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-5-%20%5Csqrt%7B13%7D%7D%7B2%20%7D)
Answer:
As the wheel makes this 270 degree counterclockwise rotation about the origin, the y-coordinate of the first car decreases from 80 to 0 and then further from 0 to -80, and finally increases to 0.
The x-coordinate decreases from 0 to -80 and then increases to 0; from there it increases further to 80.
Thus, the coordinates of the first car, after this 270-degree rotation, are (80, 0).