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
Question 2.1.1:
To find the equation of the new circle, which is the rotation of the original circle through 180 degrees around the origin, you can use the following steps:
1. Rewrite the equation of the original circle in standard form:
x^2 - 2x + y^2 - 4y - 4 = 0 becomes
(x - 1)^2 - 1 + (y - 2)^2 - 4 - 4 = 0
2. Apply the rotation transformation by switching the signs of x and y:
(y - 1)^2 - 1 + (x - 2)^2 - 4 - 4 = 0
Simplifying, you get:
(y - 1)^2 + (x - 2)^2 - 9 = 0
Therefore, the equation of the new circle, which is the rotation of the original circle through 180 degrees around the origin, is:
(y - 1)^2 + (x - 2)^2 = 9
Question 2.1.2:
To find the equation of another circle, which is the translation of the points of the original circle by (x, y) -> (x - 8, y - 6), you can use the following steps:
1. Start with the equation of the original circle:
x^2 - 2x + y^2 - 4y - 4 = 0
2. Substitute (x - 8) wherever x appears, and substitute (y - 6) wherever y appears:
(x - 8)^2 - 2(x - 8) + (y - 6)^2 - 4(y - 6) - 4 = 0
Simplify to get the equation of the new circle.
Question 2.2:
To solve question 2.2, you are given the equation of a circle: x^2 + y^2 - 8x + 16y = 15.
Question 2.2.1:
To prove whether the point (2, -9) is on the circle with the equation x^2 + y^2 - 8x + 16y = 15, substitute the coordinates (x, y) with the values (2, -9) into the equation and see if it satisfies the equation. If the equation holds true, then the point (2, -9) is on the circle.
Question 2.2.2:
To determine an equation of the tangent to the circle at the point (2, -9), you need to find the derivative of the equation of the circle and substitute the coordinates of the point (2, -9) into the derivative. The resulting equation will be the equation of the tangent to the circle at that point.
Question 2.3:
To calculate the length of the tangent AB drawn from the point A(6, 4) to the circle with the equation (x - 3)^2 + (y + 1)^2 = 10, you need to find the equation of the tangent line through point A and determine its intersection with the circle. Once you find the point of intersection, you can calculate the length of AB using the distance formula.
2.1.1 To find the equation of the new circle that is a rotation of the original circle through 180 degrees around the origin, you can use the following steps:
Step 1: Rewrite the equation of the original circle in its general form.
The equation of the original circle is x^2 - 2x + y^2 - 4y - 4 = 0.
Step 2: Apply the rotation of 180 degrees formula.
To rotate a point (x, y) by 180 degrees around the origin, you need to apply the transformation (-x, -y). Therefore, for the equation, replace x with -x and y with -y.
Step 3: Simplify the equation to get the final result.
Applying the transformation to the equation x^2 - 2x + y^2 - 4y - 4 = 0, we get (-x)^2 - 2(-x) + (-y)^2 - 4(-y) - 4 = 0. Simplifying further, the equation becomes x^2 + 2x + y^2 + 4y - 4 = 0.
So, the equation of the new circle obtained by rotating the original circle 180 degrees around the origin is x^2 + 2x + y^2 + 4y - 4 = 0.
2.1.2 To find the equation of another circle that is a translation of the points of the original circle by (x, y) = (-8, -6), follow the steps below:
Step 1: Rewrite the equation of the original circle in its general form.
The equation of the original circle is x^2 - 2x + y^2 - 4y - 4 = 0.
Step 2: Apply the translation to the equation.
To translate a point (x, y) by (x', y'), the new point becomes (x + x', y + y'). In this case, we need to subtract 8 from x and 6 from y. So, the new equation after translation becomes (x - 8)^2 - 2(x - 8) + (y - 6)^2 - 4(y - 6) - 4 = 0.
Simplifying further, we get (x - 8)^2 - 2x + 16 + (y - 6)^2 - 4y + 24 - 4 = 0.
Combining like terms, the equation simplifies to (x - 8)^2 - 2x + (y - 6)^2 - 4y + 36 = 0.
So, the equation of the new circle obtained by translating the original circle by (x, y) = (-8, -6) is (x - 8)^2 - 2x + (y - 6)^2 - 4y + 36 = 0.
2.2 To prove that the point (2, -9) is indeed on the circle with the equation x^2 + y^2 - 8x + 16y = 15, you need to substitute the coordinates of the point into the equation and check if the left-hand side equals the right-hand side.
Substituting x = 2 and y = -9 into the equation x^2 + y^2 - 8x + 16y = 15, we have:
(2)^2 + (-9)^2 - 8(2) + 16(-9) = 15
4 + 81 - 16 + (-144) = 15
85 - 160 = 15
-75 = 15
Since -75 does not equal 15, the point (2, -9) is not on the circle.
2.2.2 To determine the equation of the tangent to the circle at the point (2, -9), you need to find the derivative of the equation of the circle and use it to calculate the slope of the tangent line. Then, you can use the slope-intercept form of a line to find the equation of the tangent.
Step 1: Find the derivative of the circle equation.
Differentiating the equation x^2 + y^2 - 8x + 16y = 15 with respect to x, we have:
2x + 2yy' - 8 + 16y' = 0,
2x - 8 + 2yy' + 16y' = 0,
2(x - 4) + 2y(y' + 8) = 0.
Step 2: Find the slope of the tangent.
We know that the slope of the tangent is given by the derivative of y with respect to x. Therefore, the slope can be calculated as follows:
2(x - 4) + 2y(y' + 8) = 0,
2(x - 4) = -2y(y' + 8),
y' = -(2(x - 4))/(2y + 16),
y' = -(x - 4)/(y + 8).
At the point (2, -9), substituting x = 2 and y = -9 into the slope equation, we have:
y' = -(2 - 4)/(-9 + 8) = -(-2)/(-1) = 2.
Therefore, the slope of the tangent at the point (2, -9) is 2.
Step 3: Use the slope-intercept form to find the equation of tangent.
Using the point-slope form of a line with the slope (m = 2) and the point (2, -9), the equation of the tangent can be written as:
y - y1 = m(x - x1),
y - (-9) = 2(x - 2),
y + 9 = 2(x - 2).
Simplifying further, the equation of the tangent becomes:
y + 9 = 2x - 4,
y = 2x - 13.
So, the equation of the tangent to the circle at the point (2, -9) is y = 2x - 13.
2.3 To calculate the length of the tangent AB drawn from the point A(6, 4) to the circle with the equation (x - 3)^2 + (y + 1)^2 = 10, you can follow these steps:
Step 1: Find the equation of the line passing through point A(6, 4) and center C(3, -1) of the circle.
The equation of a line passing through two points (x1, y1) and (x2, y2) can be found using the point-slope form:
(y - y1) = m(x - x1),
(y - 4) = (4 - (-1))/(6 - 3)(x - 6).
Simplifying the equation, we get:
(y - 4) = 5/3(x - 6).
Step 2: Find the coordinates of the point B where the tangent AB intersects the circle.
Since the tangent is drawn from point A to the circle, the coordinates of the point of intersection, B, can be found by solving the system of equations formed by the line and the circle.
Substituting the equation of the line into the circle equation, we get:
(x - 3)^2 + (5/3(x - 6) + 1)^2 = 10.
Expanding and simplifying the equation, we have:
(x - 3)^2 + (5/3x - 7)^2 = 10.
Solving this equation will give the x-coordinate(s) of point B.
Step 3: Calculate the length of the tangent AB using the distance formula.
Once you have the coordinates of point B, you can calculate the distance between points A and B using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2).
Substitute the coordinates of A and B into the formula to find the length of tangent AB.