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Find the equation of a circle which touches the line 2x - y = 4 at the point (1,-2) and passes
through (3,4)
(A) (x-1)² + (y + 2)² + 10 (2x - y - 4) = 0
(B) (x - 1)² + (y + 2)² + 15 (2x - y - 4) = 0
(C) (x − 1)² + (y + 2)² + 20 (2x - y - 4) = 0
(D) (x - 1)² + (y + 2)² + 25 (2x - y - 4) = 0
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To find the equation of a circle that touches the line 2x - y = 4 at the point (1,-2) and passes through (3,4), we can first find the center of the circle and then the radius.
First, let's find the center of the circle. Since the circle touches the line at the point (1,-2), the perpendicular from the center of the circle to the line 2x - y = 4 will pass through this point.
The given line has a slope of 2, so its perpendicular will have a slope of -1/2. Using the point-slope form of a line, we can write the equation of the perpendicular line passing through (1,-2) as:
(y + 2) = (-1/2)(x - 1)
2(y + 2) = - (x - 1)
2y + 4 = -x + 1
2y = -x - 3
y = (-1/2)x - 3/2
2x - y = 4
y = (-1/2)x - 3/2
2x - (-1/2)x - 3/2 = 4
(5/2)x = 11/2
x = 11/5
= (-1/2)x - 3/2:
y = (-1/2)(11/5) - 3/2
y = -11/10 - 15/10
y = -26/10
y = -13/5
So, the intersection point of the perpendicular line and the line is (11/5, -13/5), which is the center of the circle.
Next, let's find the radius of the circle using the distance formula. The radius is the distance between the center of the circle (11/5, -13/5) and the point (3,4).
radius = sqrt((3 - 11/5)² + (4 - (-13/5))²)
radius = sqrt((15/5 - 11/5)² + (20/5 + 13/5)²)
radius = sqrt((4/5)² + (33/5)²)
radius = sqrt(16/25 + 1089/25)
radius = sqrt(1105/25)
radius = sqrt(1105) / 5
Now, we have the center of the circle (11/5, -13/5) and the radius sqrt(1105) / 5. The equation of a circle can be written as (x - h)² + (y - k)² = r², where (h,k) represents the center of the circle and r represents the radius.
(x - 11/5)² + (y + 13/5)² = (sqrt(1105) / 5)²
(x - 11/5)² + (y + 13/5)² = 1105 / 25
25(x - 11/5)² + 25(y + 13/5)² = 1105
25(x - 11/5)² + 25(y + 13/5)² - 1105 = 0
(x - 11/5)² + (y + 13/5)² - 44 = 0
(D) (x - 1)² + (y + 2)² + 25 (2x - y - 4) = 0
Verified answer
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[tex]\pink\longrightarrow[/tex]To find the equation of a circle that touches the line 2x - y = 4 at the point (1,-2) and passes through (3,4), we can first find the center of the circle and then the radius.
First, let's find the center of the circle. Since the circle touches the line at the point (1,-2), the perpendicular from the center of the circle to the line 2x - y = 4 will pass through this point.
The given line has a slope of 2, so its perpendicular will have a slope of -1/2. Using the point-slope form of a line, we can write the equation of the perpendicular line passing through (1,-2) as:
(y + 2) = (-1/2)(x - 1)
Expanding and rearranging the equation:
2(y + 2) = - (x - 1)
2y + 4 = -x + 1
2y = -x - 3
y = (-1/2)x - 3/2
Now, let's find the intersection point of the perpendicular line and the line 2x - y = 4. We can do this by solving the simultaneous equations:
2x - y = 4
y = (-1/2)x - 3/2
Substituting the equation for y in terms of x into the first equation:
2x - (-1/2)x - 3/2 = 4
(5/2)x = 11/2
x = 11/5
Substituting the x-value back into y
= (-1/2)x - 3/2:
y = (-1/2)(11/5) - 3/2
y = -11/10 - 15/10
y = -26/10
y = -13/5
So, the intersection point of the perpendicular line and the line is (11/5, -13/5), which is the center of the circle.
Next, let's find the radius of the circle using the distance formula. The radius is the distance between the center of the circle (11/5, -13/5) and the point (3,4).
Using the distance formula, we have:
radius = sqrt((3 - 11/5)² + (4 - (-13/5))²)
radius = sqrt((15/5 - 11/5)² + (20/5 + 13/5)²)
radius = sqrt((4/5)² + (33/5)²)
radius = sqrt(16/25 + 1089/25)
radius = sqrt(1105/25)
radius = sqrt(1105) / 5
Now, we have the center of the circle (11/5, -13/5) and the radius sqrt(1105) / 5. The equation of a circle can be written as (x - h)² + (y - k)² = r², where (h,k) represents the center of the circle and r represents the radius.
Substituting the values, we get:
(x - 11/5)² + (y + 13/5)² = (sqrt(1105) / 5)²
(x - 11/5)² + (y + 13/5)² = 1105 / 25
Multiplying both sides by 25 to eliminate the fractions, we obtain:
25(x - 11/5)² + 25(y + 13/5)² = 1105
Expanding and rearranging further, we get:
25(x - 11/5)² + 25(y + 13/5)² - 1105 = 0
(x - 11/5)² + (y + 13/5)² - 44 = 0
Comparing this equation with the options provided, we see that the correct answer is:
(D) (x - 1)² + (y + 2)² + 25 (2x - y - 4)=0
[tex]{ \orange{ \sf \:hope \: it \: helps \: you \: }}[/tex]