Answer:
Equation of the circle
(x - 0.7)² + (y - 1.3)² = 12.58
Step-by-step explanation:
The formula for the equation of a circle is given as:
(x - a)² + (y - b)² = r²,
where(a, b) is the center of the circle and r = radius of the circle.
a) We are told in the question that the equation of the circle passes through point(2, -2)
Hence,
Substituting 2 for x and -2 for y in the equation of the circle.
(x - a)² + (y - b)² = r²
(2 - a)² +(-2 - b)² = r²
Expanding the bracket
(2 - a) (2 - a) + (-2 - b)(-2 - b) = r²
4 - 2a - 2a +a² +4 +2b +2b +b² = r²
4 - 4a + a² + 4 + 4b + b² = r²
a² + b² -4a + 4b + 4 + 4 = r²
a² + b² -4a + 4b + 8 = r²............Equation 1
We are also told that the equation of the circle also passes through point (3,4) also, where 3 = x and 4 = y
Hence,
Substituting 3 for x and 4 for y in the equation of the circle.
(x - a)² + (y - b)² = r²
(3 - a)² +(4 - b)² = r²
Expanding the bracket
(3 - a) (3 - a) + (4 - b)(4- b) = r²
9 - 3a - 3a +a² +16 -4b -4b +b² = r²
9 -6a + a² + 16 -8b + b² = r²
a² + b² -6a -8b + 9 + 16 = r²
a² + b² -6a -8b + 25 = r²..........Equation 2
The next step would be to subtract Equation 1 from Equation 2
a² + b² -4a + 4b + 8 - (a² + b² -6a -8b + 25) = r² - r²
a² + b² -4a + 4b + 8 - a² - b² +6a +8b - -25= r² - r²
Collecting like terms
a² - a² + b² - b² - 4a + 6a + 4b + 8b +8- 25 = 0
2a + 12b -17 = 0
2a + 12b = 17...........Equation 3
Step 2
We are going to have to find the values of a and b in other to get our equation of the circle.
Since the center of the circle(a, b) lies on x + y = 2
Therefore, we have
a + b = 2
a = 2 - b
2a + 12b = 17 ..........Equation 3
Substituting 2 - b for a in
2(2 - b) + 12b = 17
4 - 2b + 12b = 17
4 + 10b = 17
10b = 17 - 4
10b = 13
b = 13/10
b = 1.3
Substituting 1.3 for b in
a + b = 2
a + 1.3 = 2
a = 2 - 1.3
a = 0.7
hence, a = 0.7, b = 1.3
Step 3
We have to find the value of r using points (2, -2)
(x - a)² + (y - b)² = r²
Where x = 2 and y = -2
(-2 - 0.7)² + (-2 - 1.3)² = r²
(-2.7)² + (-3.3)² = r²
1.69 + 10.89 = r²
r² = 12.58
r = √12.58 = 3.55
Step 4
The formula for the equation of a circle is given as:
(x - a)² + (y - b)² = r²,
where(a, b) is the center of the circle and r = radius of the circle
a = 0.7
b = 1.3
r² = 12.58
Equation of the circle =
(x - 0.7)² + (y - 1.3)² = 12.58
. The third solution can be easily found using the rational root theorem, and it's equal to -1.