Conic Sections: Circles and Ellipses
Writing equations and analyzing key features
Conic Sections: Circles and Ellipses
Writing equations and analyzing key features
Math - Grade 9-12
- 1
Write the standard form equation of a circle with center (2, -3) and radius 5.
Use the circle formula with h = 2, k = -3, and r = 5.
The standard form equation is (x - 2)^2 + (y + 3)^2 = 25. A circle with center (h, k) and radius r has equation (x - h)^2 + (y - k)^2 = r^2. - 2
Find the center and radius of the circle x^2 + y^2 - 6x + 10y + 9 = 0.
The center is (3, -5) and the radius is 5. Completing the square gives (x - 3)^2 + (y + 5)^2 = 25. - 3
Determine whether the equation (x + 1)^2 + (y - 4)^2 = 49 represents a circle. If it does, state its center and radius.
Compare the equation to the standard form of a circle.
Yes, the equation represents a circle. Its center is (-1, 4) and its radius is 7 because 49 = 7^2. - 4
Write the equation of a circle whose diameter has endpoints (-2, 1) and (6, 1).
The center is the midpoint, which is (2, 1). The diameter is 8, so the radius is 4. The equation is (x - 2)^2 + (y - 1)^2 = 16. - 5
A circle has center (0, 0) and passes through the point (8, 6). Write its equation.
Use the distance formula to find the radius.
The radius is the distance from (0, 0) to (8, 6), which is sqrt(8^2 + 6^2) = 10. The equation is x^2 + y^2 = 100. - 6
State the center, vertices, co-vertices, and foci of the ellipse (x - 3)^2/25 + (y + 2)^2/9 = 1.
The center is (3, -2). Since 25 is larger than 9, the major axis is horizontal with a = 5 and b = 3. The vertices are (8, -2) and (-2, -2). The co-vertices are (3, 1) and (3, -5). The foci are (7, -2) and (-1, -2) because c = sqrt(25 - 9) = 4. - 7
Write the standard form equation of an ellipse with center (0, 0), horizontal major axis, a = 6, and b = 4.
The equation is x^2/36 + y^2/16 = 1. For a horizontal ellipse centered at the origin, the larger denominator is under x. - 8
Find the center and lengths of the major and minor axes of the ellipse x^2/49 + y^2/9 = 1.
Axis lengths are 2a and 2b.
The center is (0, 0). Since 49 is larger than 9, a = 7 and b = 3. The major axis length is 14 and the minor axis length is 6. - 9
Determine whether the ellipse (x + 2)^2/16 + (y - 5)^2/36 = 1 has a horizontal or vertical major axis. Then state its vertices.
The major axis is vertical because 36 is the larger denominator and it is under the y-term. The center is (-2, 5), so the vertices are (-2, 11) and (-2, -1). - 10
For the ellipse x^2/64 + y^2/48 = 1, find the value of c and the coordinates of the foci.
Use the relationship c^2 = a^2 - b^2.
Here a^2 = 64 and b^2 = 48, so c^2 = 64 - 48 = 16 and c = 4. The major axis is horizontal, so the foci are (4, 0) and (-4, 0). - 11
Write the equation of an ellipse centered at (1, -2) with vertical major axis, a = 7, and b = 3.
The equation is (x - 1)^2/9 + (y + 2)^2/49 = 1. For a vertical major axis, the larger denominator is under the y-term. - 12
Complete the square to rewrite 4x^2 + 4y^2 - 16x + 8y - 20 = 0 in standard form. Then identify the graph.
Divide by 4 before completing the square.
First divide by 4 to get x^2 + y^2 - 4x + 2y - 5 = 0. Then rearrange and complete the square: (x - 2)^2 + (y + 1)^2 = 10. The graph is a circle with center (2, -1) and radius sqrt(10). - 13
An ellipse has center (0, 0), vertices at (0, 9) and (0, -9), and co-vertices at (4, 0) and (-4, 0). Write its equation.
The major axis is vertical, so a = 9 and b = 4. The equation is x^2/16 + y^2/81 = 1. - 14
A circle is tangent to the x-axis and has center (3, 5). Write the equation of the circle.
The radius equals the vertical distance from the center to the x-axis.
Because the circle is tangent to the x-axis, its radius is the distance from the center to the x-axis, which is 5. The equation is (x - 3)^2 + (y - 5)^2 = 25. - 15
Compare the equations x^2/25 + y^2/25 = 1 and x^2/25 + y^2/9 = 1. Identify which graph is a circle and which is an ellipse, and explain why.
The equation x^2/25 + y^2/25 = 1 is a circle because both denominators are equal, so the distance from the center is the same in every direction. The equation x^2/25 + y^2/9 = 1 is an ellipse because the denominators are different, so the horizontal and vertical stretches are not equal.