# Lesson 10-1: Distance and Midpoint

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Lesson 10-1: Distance and Midpoint

Distance Formula Midpoint Formula

Find distance and midpoint
(0, 0) (1, -4)

(2, 4) (-5, -1)

Two cities are located on a map using a coordinate system
Two cities are located on a map using a coordinate system. Your house is exactly half-way between the two cities. If city #1 is located at (-12, 2) and your house is at (-7.75, -4.5). What is the grid location of city #2?

A circle has diameter 𝐴𝐵
A circle has diameter 𝐴𝐵 . If A is at (-3,-5) and the center of the circle is at (2, 3), find the coordinates of B. Then find the circumference and area of the circle.

Find the perimeter of a triangle with vertices of A(4, 1), B(-3, -2), and C(-1, -4).

Lesson 10-2: Parabolas

Conic section: Any figure that can be obtained by slicing a double cone
Focus: the point that is the same distance from all points in a parabola Directrix: a given line that is the same distance from all points in a parabola Latus rectum: the line segment through the focus of a parabola and perpendicular to the axis of symmetry

Upward if a>0, downward if a<0
Parabolas- y = a(x – h)2 + k x = a(y – k)2 + h Vertex (h, k) Axis of symmetry x = h y = k Focus (h, k 𝑎 ) (h 𝑎 , k) Directrix y = k 𝑎 x = h 𝑎 Direction of Opening Upward if a>0, downward if a<0 Right if a>0, Left if a<0 Length of Latus rectum 1 𝑎 units

Write the equation in standard form
Write the equation in standard form. Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening. y = x2 – 6x + 11

Write the equation in standard form
Write the equation in standard form. Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening. x = 3y2 + 5y - 9

Vertex (8, 6) focus (2, 6) Vertex (3, 4) axis of symmetry x = 3, measure of latus rectum 4, a>0

Vertex (1, 7) directrix y = 3

Graph.

Graph.

Lesson 10-3: Circles

Equation of a circle (x – h)2 + (y – k)2 = r2
Circle: the set of all points in a plane that are equidistant from a given point in the plane Center: the point that all points in a circle are equidistant from Equation of a circle (x – h)2 + (y – k)2 = r2 h = x value of center k = y value of center r = radius length .

Graph (not in packet) Center (8, -3) r=6

Identify the center and radius for each circle given
Identify the center and radius for each circle given. Then graph the circle. Center (7, -3) passes through the origin

Center (-2, 8) and tangent to y=4

(x-3)2 + y2 = 9

Write the equation in standard form then graph.
x2 + y2 – 4x + 8y – 5 = 0

Write the equation in standard form then graph.
x2 + y2 + 4x - 10y – 7 = 0

Write the equation for the circle described.
Center (-1,-5) radius 2 units Endpoints of a diameter at (-4, 1) and (4, -5)

A plan for a park puts the center of a circular pond of radius 0
A plan for a park puts the center of a circular pond of radius 0.6mi, 2.5mi east and 3.8mi south of the park headquarters. Use the headquarters as the origin and write an equation to represent the situation.

Lesson 10.4: Ellipses

Ellipse: the set of all points in a plane such that the sum of the distance from two fixed points is constant Foci: the two fixed points of an ellipse Major axis: the longer line segment that forms an axis of symmetry for an ellipse Minor axis: the shorter line segment that forms an axis of symmetry for an ellipse Center: the intersection of the axes of symmetry for an ellipse

Direction of major axis
(𝒙−𝒉) 𝟐 𝒂 𝟐 + (𝒚−𝒌) 𝟐 𝒃 𝟐 =𝟏 (𝒚 −𝒌) 𝟐 𝒂 𝟐 + (𝒙 −𝒉) 𝟐 𝒃 𝟐 =𝟏 Direction of major axis horizontal vertical Foci (h + c, k) and (h - c, k) (h, k + c) and (h, k - c) Length of major axis 2a units Length of minor axis 2b units Center is (h , k) and NOTE:

State the center, the direction of the major axis, the length of the major and minor axis, the value of c, and the foci. 𝑥 𝑦 2 4 =1 𝑥 𝑦 =1

𝑥 𝑦 =1 (𝑦 −3) (𝑥 −2) =1

Write an equation for the ellipse described.
Endpoints of the major axis at (-5, 0) and (5, 0). Endpoints of the minor axis at (0, -2) and (0, 2).

Write an equation for the ellipse described.
Major axis is 20 units long and parallel to y-axis Minor axis is 6 units long and center at (4, 2)

Write the equation in standard form.
7x2 + 3y2 – 28x – 12y = -19

Write the equation for each ellipse in standard form, then state the center, the foci, the length of the major and minor axes. Then graph the ellipse. x2 + 4y2 +4x – 24y + 24 = 0

Write the equation for each ellipse in standard form, then state the center, the foci, the length of the major and minor axes. Then graph the ellipse. 3x2 + y2 = 9

Write the equation for each ellipse in standard form, then state the center, the foci, the length of the major and minor axes. Then graph the ellipse. 4x2 + 3y2 = 48

Lesson 10-5: Hyperbolas

Hyperbola: the set of all points in a plane such that the absolute value of the differences of the distances from two fixed points is constant Center: intersection of transverse and conjugate axes Transverse axis: axis of symmetry whose endpoints are the vertices of the hyperbola Conjugate axis: axis of symmetry perpendicular to the transverse axis

𝒙 − 𝒉 𝟐 𝒂 𝟐 − 𝒚 −𝒌 𝟐 𝒃 𝟐 =𝟏 𝒚 −𝒌 𝟐 𝒂 𝟐 − 𝒙 −𝒉 𝟐 𝒃 𝟐 =𝟏
𝒙 − 𝒉 𝟐 𝒂 𝟐 − 𝒚 −𝒌 𝟐 𝒃 𝟐 =𝟏 𝒚 −𝒌 𝟐 𝒂 𝟐 − 𝒙 −𝒉 𝟐 𝒃 𝟐 =𝟏 Direction of transverse axis horizontal vertical Foci (h ± c, k) (h, k ± c) Vertices (h ± a, k) (h, k ± a) Length of transverse axis 2a units Length of conjugate axis 2b units Asymptotes y – k = ± 𝑏 𝑎 (x – h) y – k = ± 𝑎 𝑏 (x – h) *Note: Center is (h , k) and c2 = a2 + b2

Write the equation for the hyperbola.

Vertices (-5, 0) and conjugate axis length 12 units
Foci (-4, 5 ± 97 )

Find the coordinates of the vertices and foci and the equations of the aysmptotes. Then graph the hyperbola. 𝑦 − 𝑥 2 7 =1

Find the coordinates of the vertices and foci and the equations of the aysmptotes. Then graph the hyperbola. 𝑥 − 𝑦 =1

Find the coordinates of the vertices and foci and the equations of the aysmptotes. Then graph the hyperbola. 𝑦 −3 2 − 𝑥 =1

Find the coordinates of the vertices and foci and the equations of the aysmptotes. Then graph the hyperbola. 4x2 – 25y2 - 8x – 96 = 0

10.6 Conic Sections

Standard Form of Equation
Conic Section Standard Form of Equation Parabola y = a(x - h)2 + k or x = a(y – k)2 + h Circle (x – h)2 + (y – k)2 = r2 Ellipse 𝑥 −ℎ 2 𝑎 𝑦 −𝑘 2 𝑏 2 =1 or 𝑦 −𝑘 2 𝑎 𝑥 −ℎ 2 𝑏 2 =1 Hyperbola 𝑥 −ℎ 2 𝑎 2 − 𝑦 −𝑘 2 𝑏 2 =1 or 𝑦 −𝑘 2 𝑎 2 − 𝑥 −ℎ 2 𝑏 2 =1

Write the equation in standard form
Write the equation in standard form. State whether it is a parabola, circle, ellipse, or hyperbola. Then graph. x2 + 4y2 – 6x – 7 = 0

y = x2 + 3x + 1

A and C have the same sign and A ≠ C Hyperbola
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 Parabola A = 0 or C = o but not both Circle A = C Ellipse A and C have the same sign and A ≠ C Hyperbola A and C have opposite signs

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. y2 – x – 10y + 34 = 0 y2 – 2x2 - 4x – 4y – 4 = 0 3x2 + 2y x – 28y = 0 4x2 + 4y x – 12y + 30 = 0

A military jet performs for an air show
A military jet performs for an air show. The path of the plane during one trick can be modeled by a conic section with equation 24x y – 31,680x – 45,600 = 0. Distances are represented in feet. Identify the shape of the curved path of the jet. Write the equation in standard form.

If the jet begins its path upward or ascent at (0, 0), what is the horizontal distance traveled by the jet from the beginning of the ascent to the end of the descent?

What is the maximum height of the jet?

Review how to solve a system
Elimination: If two coefficients are the same add or subtract to cancel that variable If needed, multiply to get like coefficients and then add or subtract Substitution: Solve one of the equations for a variable and then replace that variable in the other equation to solve.

x2-4y2=9 4y-x=3

y=x-1 x2+y2=25

x+y=1 y=x2+5

y2=13-x2 x2+4y=25

x2+y2=36 x2+9y2=36

Solving Systems Graph both inequalities and test a point inside the conic section to see where you are to shade The shaded part that overlaps is your solution.

y≤x2-2 x2+y2<16

y>x2+1 x2+y2≤9

x2+y2≤49 y≥x2+1