Algebra 2 Page 28 of 84

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Analytic Geometry and Conic Sections

1. Distance and Midpoint Formulas

2. Parabolas

3. Circles

4. Ellipses

5. Hyperbolas

Algebra 2 Page 29 of 84

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1. Distance and Midpoint Formulas

1) The distance formula,

2) The midpoint formula,

3) Problems (A-Plus Notes for Algebra, p257)

①. Find the distance and midpoint between and .

, (1.5, 2.5)

②. Find the distance and midpoint between and .

,

③. If M (1,0) is the midpoint of the segment and is the coordinates of Q.

Find the coordinates of P.

④. Find the distance between two points with the given coordinates. (1,3), (2,6)

⑤. Find the distance between two points with the given coordinates. (3,1), (6,2)

⑥. Find the distance between two points with the given coordinates. (-1,3), (2,-6)

⑦. Find the distance between two points with the given coordinates. (-1,-3), (-2,6)

⑧. Find the distance between two points with the given coordinates. (1,-3), (-2,-6)

⑨. Find the distance between two points with the given coordinates. (-1,-3), (-2,-6)

⑩. Find the distance between two points with the given coordinates. (5,0), (0,7)

⑪. Find the distance between two points with the given coordinates. (5,0), (7,0)

⑫. Find the distance between two points with the given coordinates. (5,7), (0,0)

⑬. Find the distance between two points with the given coordinates. (0.5,-3), (0.8,2)

⑭. Find the distance between two points with the given coordinates.

⑮. Find the distance between two points with the given coordinates.

⑯. Find the distance between two points with the given coordinates.

⑰. Find the distance between two points with the given coordinates.

⑱. Find the distance between two points with the given coordinates.

⑲. Find the distance between two points with the given coordinates.

⑳. Find the distance between two points with the given coordinates.

21. Find the distance between two points with the given coordinates.

22. Find the distance between two points with the given coordinates.

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23. Find the distance between two points with the given coordinates.

24. Find the midpoint of the line segment between two points.

25. Find the midpoint of the line segment between two points.

26. Find the midpoint of the line segment between two points.

27. Find the midpoint of the line segment between two points.

28. Find the midpoint of the line segment between two points.

29. Find the midpoint of the line segment between two points.

30. Find the midpoint of the line segment between two points.

31. Find the midpoint of the line segment between two points.

32. Find the midpoint of the line segment between two points.

33. Find the midpoint of the line segment between two points.

34. Find the midpoint of the line segment between two points.

35. Find the midpoint of the line segment between two points.

36. Find the midpoint of the line segment between two points.

37. Find the midpoint of the line segment between two points.

38. Find the midpoint of the line segment between two points.

39. If is the midpoint of the line segment and is the coordinates of

one endpoint, find the coordinates of the other endpoint.

40. If is the midpoint of the line segment and is the coordinates of

one endpoint, find the coordinates of the other endpoint.

41. The line segment is the diameter of a circle. If the center is at and A is at

, find the coordinates of B.

42. Find the equation of the line that passes (3,-2) and is perpendicular to the line .

43. Find the equation of the line that is perpendicular bisector of the line segment between

two points (5,-1) and (3,7).

44. The vertices of a triangle are (3,3), (-3,3) and (0,-1). Identify the triangle as scalene,

isosceles, or equilateral.

Algebra 2 Page 31 of 84

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2. Parabolas

1) Definition of a Parabola: A parabola is the set of all points (x, y) in a plane that are equidistant

from a fixed line, the directrix, and a fixed point, the focus, not on the line. The vertex is the

midpoint between the focus and the directrix. The axis of the parabola is the line passing through

the focus and the vertex.

2) Standard Equation of a Parabola (Vertex at Origin) and

directrix , the equation is

directrix , the equation is

The focus is on the axis p units (directed distance) from the vertex.

3) Problems (A-Plus Notes for Algebra, p259)

①. Find the equation of a parabola whose focus is the point (0,2) and whose directrix is the

line .

②. Find the equation fo a parabola whose focus is the point (0,3) and whose directrix is the

line .

③. Find an equation of the parabola with vertex (0,0) and focus (2,0).

④. Find an equation of the parabola with vertex (0,0) and focus (-2,0).

⑤. Find the equation of a parabola whose focus is the point (0,4) and whose directrix is the

line .

⑥. Find the equation of a parabola whose focus is the point (0,2) and whose directrix is the

line .

⑦. Find the vertex, focus, directrix of the parabola .

y

x

Vertex

Focus

Directrix

Algebra 2 Page 32 of 84

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3. Circles

1) Definition: A circle is the set of all points in a plane that are a fixed distance r (radius) from a

fixed point (center).

2) Standard forms of circles

①. Center (0,0) and radius r:

②. Center (a, b) and radius r:

3) Problems

①. Find the center and radius of .

②. Find the equation of the circle with center (0,0) and radius .

③. Find the equation of the circle with center (-2,1) and radius 3.

④. Draw the graph .

⑤. Write the equation of a circle if the endpoints of the diameter are at (5,4) and (-1,-2).

⑥. Write the standard form of the equation of each circle. r = 1 and Center (0,0)

⑦. Write the standard form of the equation of each circle. r = 1 and Center (0,5)

⑧. Write the standard form of the equation of each circle. r = 1 and Center (-5,0)

⑨. Write the standard form of the equation of each circle. r = 1 and Center (3,5)

⑩. Write the standard form of the equation of each circle. r = 3 and Center (5,-3)

⑪. Write the standard form of the equation of each circle. r = 5 and Center (-3,5)

⑫. Write the standard form of the equation of each circle. r = 8 and Center (-1,-6)

⑬. Write the standard form of the equation of each circle. r = and Center (4,-2)

⑭. Write the standard form of the equation of each circle. r = and Center ( ,0)

⑮. Find the center and radius of each circle.

Algebra 2 Page 33 of 84

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4. Ellipses

1) Definition: An ellipse is the set of all points in a plane such that for each point, the sum of

the distances (the focal radii) from two fixed points (foci) is a constant.

2) Standard forms of ellipses

①. Center (0,0), focus : ,

②. Center (h, k), focus : ,

③. Center (0, 0), focus : ,

④. Center (h, k), focus : ,

3) Problems

①. Find the equation of an ellipse having foci (0, 3) and (0, -3) and the sum of its focal

radii is 10.

②. Draw the graph .

③. Draw the graph .

④. Draw the graph .

⑤. Find the equation of an ellipse having vertices at (0, 5) and (0, -5) and foci at (0, 3)

and (0, -3).

⑥. Find the equation of an ellipse passing through the point (3, 5) and having the foci at

(3, 2) and (-1, 2).

⑦. The earth’s orbit around the sum is shaped as an ellipse in which the sun is located

and fixed at one of its two foci. At its closest distance, the earth is 91.45 million

miles from the center of the sun. At its farthest distance, the earth is 94.55 million

miles from the center of the sun. Write the equation of the orbit.

⑧. Find the center, vertices on the major axis, and foci of each ellipse. Graph each

equation.

Algebra 2 Page 34 of 84

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⑨. Write the equation in standard form for each ellipse having the given conditions.

Center (0, 0), Vertex (5, 0), Co-vertex (0, 2)

Center (0, 0), Vertex (0, -5), Co-vertex (2, 0)

Center (0, 0), Vertex (0, 8), Focus (0, 5)

Center (0, 0), Co-vertex (0, 1), Focus (2, 0)

Center (2, -1), Vertex (8, -1), Focus (6, -1)

Length of major axis: 10, Foci (6, 1) and (-2, 1)

Center (1, 2), Vertex (1, 4), Passing through the point (2, 2)

Algebra 2 Page 35 of 84

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5. Hyperbolas

1) Definition: A hyperbola is the set of all points in a plane such that for each point, the

difference of the distances from two fixed points is a constant.

2) Standard forms of hyperbolas

①. Center (0, 0), :

②. Center (h, k), :

③. Center (0, 0), :

④. Center (h, k), :

3) Problems

①. Find the equation of a hyperbola having foci (0, 3) and (0, -3) and the difference of

its focal radii is 4.

②. Draw graph .

③. Draw graph .

④. Draw graph .

⑤. Find the equation of a hyperbola having vertices at (0, 2) and (0, -2) and foci (0, 3)

and (0, -3).

⑥. Find the equation of a hyperbola passing through the point (1, -5) and having the foci

at (-2, -1) and (-2, -5).

⑦. Find the equations of the asymptotes of the hyperbola .

⑧. Find the equations of the asymptotes of the hyperbola .

⑨. Find the equations of the asymptotes of the hyperbola

.

⑩. Find the equations of the asymptotes of the hyperbola .

⑪. Find the center, vertices, and foci of each hyperbola. Graph each equation.

Algebra 2 Page 36 of 84

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⑫. Write the equation in standard form for each hyperbola having the given conditions.

Center (0, 0), Vertex (5, 0), Focus (6, 0)

Center (0, 0), Vertex (0, -5), Focus (0, 6)

Center (0, 0), Vertex (0, 5), Focus (0, 8)

Center (0, 0), Vertex (0, 1), Focus (0, 2)

Center (2, -1), Vertex (8, -1), Focus (9, -1)

Center (-2, 1), Foci (7, 1) and (-3, 1)

Vertex (0, 4) and (0, -4), Asymptote .

Vertices (1, -3) and (1, 1), Asymptote .

Foci (3, 0) and (-3, 0), Passing the point (5, 4)

Algebra 2 Page 37 of 84

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Parabola

Same distance from a point (focus) and a line (directrix)

, the equation is

, the equation is

Circle

Same distance from a point (center)

Center (0,0) and radius r:

Center (a, b) and radius r:

Ellipse

The sum of distance from two points (foci) is constant

Center (0,0), focus : ,

Center (h, k), focus : ,

Center (0, 0), focus : ,

Center (h, k), focus : ,

Hyperbola

The difference of the distance from two points (foci) is constant

Center (0, 0), :

Center (h, k), :

Center (0, 0), :

Center (h, k), :