find the equation of circle
if the line 2x - y +1 = 0 touches the circle at the point (2,5) and the center of the circle lies on the line x+y -9 =0
If the line touches the circle, then the line from (2,5) to the center is perpendicular to the line y=2x+1, so it has slope -1/2.
So, the line through (2,5) with slope -1/2 is
y-5 = -1/2 (x-2)
y = -1/2 x + 6
This line passes through the center of the circle, so
-1/2 x + 6 = 9-x
x=6
(6,3) is the center of the circle.
The radius is the distance from (6,3) to (2,5) = √20
The circle is thus
(x-6)^2 + (y-3)^2 = 20
Just as a final check:
The distance from (6,3) to the line 2x-y+1 = 0 is |2*6 - 3 + 1|/√5 = 10/√5 = 2√5 = √20
To find the equation of the circle, we need to determine the center and radius.
Step 1: Find the center of the circle
Since the center of the circle lies on the line x + y - 9 = 0, we can express y in terms of x as y = 9 - x.
Now substitute this value of y into the equation of the line 2x - y + 1 = 0 to find the x-coordinate of the center of the circle.
2x - (9 - x) + 1 = 0
2x - 9 + x + 1 = 0
3x - 8 = 0
3x = 8
x = 8/3
Substitute this value of x back into the equation y = 9 - x to find the y-coordinate of the center.
y = 9 - 8/3
y = 27/3 - 8/3
y = 19/3
So, the center of the circle is (8/3, 19/3).
Step 2: Find the radius of the circle
The distance between the center of the circle and the point (2, 5) is equal to the radius of the circle.
Using the distance formula:
Distance = √[(x2 - x1)² + (y2 - y1)²]
= √[(2 - 8/3)² + (5 - 19/3)²]
= √[(-14/3)² + (-4/3)²]
= √[(196/9) + (16/9)]
= √(212/9)
= √212/3
Therefore, the radius of the circle is √212/3.
Step 3: Write the equation of the circle
The standard equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
Plugging in the values we found:
(x - 8/3)² + (y - 19/3)² = (√212/3)²
Hence, the equation of the circle is (x - 8/3)² + (y - 19/3)² = 212/9.
To find the equation of a circle, a few steps are involved:
Step 1: Find the center of the circle.
Given that the center lies on the line x+y-9=0, we can rewrite it in the standard form of equation Ax + By + C = 0.
x + y - 9 = 0 can be rewritten as x + y = 9.
Now, any point that satisfies this equation will lie on the line.
Step 2: Determine the radius of the circle.
Since the line 2x - y + 1 = 0 touches the circle at the point (2,5), we can find the distance between the center and the point to obtain the radius of the circle.
Step 3: Write the equation of the circle.
Using the center coordinates from Step 1 and the radius from Step 2, we can write the equation of the circle in the standard form (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r represents the radius.
Now, let's work through the steps:
Step 1: Finding the center of the circle:
We have the equation x + y = 9. To find the center, we solve this equation and determine both x and y.
By rearranging the equation, we get y = 9 - x.
Substituting x = 9 - y into the equation, we find the value of y:
y = 9 - (9 - y)
y = 9 - 9 + y
y = y
We can substitute this value of y back into the equation to find x:
x + y = 9
x + y = 9
x + y - y = 9 - y
x = 9 - y
Therefore, the center of the circle is (9-y, y).
Step 2: Determining the radius of the circle:
To find the radius, we simply calculate the distance between the center of the circle (9-y, y) and the point where the line 2x - y + 1 = 0 touches the circle, which is (2, 5).
The distance formula is given by: √((x2 - x1)² + (y2 - y1)²).
Using the formula, we can calculate the radius as follows:
r = √((2 - (9 - y))² + (5 - y)²)
r = √((2 - 9 + y)² + (5 - y)²)
r = √((-7 + y)² + (5 - y)²)
r = √((49 - 14y + y²) + (25 - 10y + y²))
r = √(74 - 24y + 2y²)
Step 3: Writing the equation of the circle:
Using the center (9-y, y) from Step 1 and the radius expression from Step 2, the equation of the circle is:
(x - (9-y))² + (y - y)² = (√(74 - 24y + 2y²))²
Simplifying further, we get:
(x - 9 + y)² + y² = 74 - 24y + 2y²
And that is the equation of the circle.