The circle x2-2x+y2-4y-4=0 lies on the cartesian plane,question 2.1.1 equation of new circle which is the rotation of original circle through 180 degree around the origin...2.1.2 find equation of another circle which is the translation (x;y)-(x-8;y-6) of the points of the original circle..2.2 equation of a circle is x2+y2-8x+16y=15...2.2.1 prove that the point (2;-9) is on the circle...2.2.2 determine an equation of the tangent to the circle at the point (2;-9)...2.3 calculate the length of the tangent AB drawn from the point A(6;4) to the circle with equation (x-3)squared+(y+1)squared=10
2.1.1 To find the equation of the new circle after rotating the original circle 180 degrees around the origin:
Step 1: Rewrite the original circle equation in terms of the origin as the center:
(x - 1)² + (y - 2)² = 4
Step 2: Apply the rotation transformation by replacing x with -x and y with -y:
(-x - 1)² + (-y - 2)² = 4
Simplify:
(x + 1)² + (y + 2)² = 4
The equation of the new circle, after rotating the original circle 180 degrees around the origin, is (x + 1)² + (y + 2)² = 4.
2.1.2 To find the equation of another circle obtained by translating the original circle according to the transformation (x, y) -> (x - 8, y - 6):
Step 1: Rewrite the original circle equation in expanded form:
x² - 2x + y² - 4y - 4 = 0
Step 2: Apply the translation by replacing x with (x - 8) and y with (y - 6):
(x - 8)² - 2(x - 8) + (y - 6)² - 4(y - 6) - 4 = 0
Expand and simplify:
x² - 16x + 64 - 2x + 16 + y² - 12y + 36 - 4y + 24 - 4 = 0
x² - 18x + y² - 16y + 84 = 0
The equation of the new circle, obtained by translating the original circle according to the transformation (x, y) -> (x - 8, y - 6), is x² - 18x + y² - 16y + 84 = 0.
2.2 To prove that the point (2, -9) is on the circle with equation x² + y² - 8x + 16y = 15:
Step 1: Substitute the given coordinates of the point (2, -9) into the equation:
(2)² + (-9)² - 8(2) + 16(-9) = 15
Simplify:
4 + 81 - 16 + (-144) = 15
85 - 160 = 15
-75 = 15
Since -75 is not equal to 15, the point (2, -9) is not on the circle with equation x² + y² - 8x + 16y = 15.
2.2.2 To determine the equation of the tangent to the circle at the point (2, -9):
Step 1: Rewrite the circle equation in completed square form:
x² - 8x + y² + 16y = 15
(x² - 8x + 16) + (y² + 16y + 64) = 15 + 16 + 64
(x - 4)² + (y + 8)² = 95
Step 2: Differentiate both sides of the equation with respect to x to find the derivative:
2(x - 4) + 2(y + 8) * dy/dx = 0
Step 3: Substitute the x-coordinate of the given point (2, -9) into the derived equation:
2(2 - 4) + 2(-9 + 8) * dy/dx = 0
-4 - 2 * dy/dx = 0
dy/dx = -2/-2
dy/dx = 1
The derivative dy/dx at the given point (2, -9) is equal to 1.
Step 4: Use the point-slope form of a line to determine the equation of the tangent:
(y - y1) = m(x - x1)
(y - (-9)) = 1(x - 2)
y + 9 = x - 2
y = x - 11
Therefore, the equation of the tangent to the circle at the point (2, -9) is y = x - 11.
2.3 To calculate the length of the tangent AB drawn from the point A(6, 4) to the circle with equation (x - 3)² + (y + 1)² = 10:
Step 1: Calculate the distance between the center of the circle (3, -1) and the point A(6, 4):
Distance = √((x2 - x1)² + (y2 - y1)²)
Distance = √((6 - 3)² + (4 - (-1))²)
Distance = √(3² + 5²)
Distance = √(9 + 25)
Distance = √34
Step 2: Calculate the radius of the circle by taking the square root of 10.
Radius = √10
Step 3: Calculate the length of the tangent AB using the Pythagorean theorem:
Length of Tangent AB = √(Distance² - Radius²)
Length of Tangent AB = √(34 - 10)
Length of Tangent AB = √24
Length of Tangent AB = 2√6
Therefore, the length of the tangent AB drawn from the point A(6, 4) to the circle with equation (x - 3)² + (y + 1)² = 10 is 2√6.