The length of CE is 3.6 and the radius of the circle is 2
The length of CE
Start by calculating the length of AD using:
AD^2 = AC^2 - CD^2
This gives
AD^2 = 10^2 - 6^2
AD^2 = 64
Take the square roots
AD = 8
Represent CE with x.
So:
AE = 10 - x
Calculate ED using:
ED^2 = CD^2 - CE^2
ED^2 = AD^2 - AE^2
So, we have:
AD^2 - AE^2 = CD^2 - CE^2
This gives
8^2 - (10 - x)^2 = 6^2 - x^2
Expand
64 - 100 + 20x - x^2 = 36 - x^2
Add x^2 to both sides
64 - 100 + 20x = 36
Evaluate the like terms
20x = 72
Divide by 20
x = 3.6
Hence, the length of CE is 3.6
The radius of the circle
The equation is given as:
x^2 + y^2 + 6x + 5y = -45/4
Rewrite as:
x^2 + 6x + y^2 + 5y = -45/4
Take the coefficients of x and y
Divide them by 2, square the result and add the result to both sides.
So, we have:
x^2 + 6x + (6/2)^2 + y^2 + 5y + (5/2)^2= -45/4+ (6/2)^2 + (5/2)^2
This gives
x^2 + 6x + (6/2)^2 + y^2 + 5y + (5/2)^2= 4
Express as perfect squares
(x + 3)^2 + (y + 5/2)^2 = 4
The circle equation is represented as:
(x - a)^2 + (y - b)^2 = r^2
Where r represents the radius
By comparison, we have:
r^2 = 4
This gives
r = 2
Hence, the radius of the circle is 2
Read more about circle equation at:
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