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SECTION: 10 – 3 HYPERBOLAS WARM-UP Find the center, vertices, foci, and eccentricity of each ellipse.

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HYPERBOLA. A hyperbola is the set of all points (x,y) the difference of whose distances (d1,d2) from two distinct fixed points (foci) is a positive constant. DESCRIPTION BRANCHES. Every hyperbola has two disconnected branches, which form the curve of the hyperbola. TRANSVERSE AXIS. The transverse axis is the line segment passing through the two foci and the two vertices. CENTER. The midpoint of the transverse axis is the center of the hyperbola.

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**STANDARD FORM EQUATION OF A HYPERBOLA**

STANDARD FORM EQUATION OF A HYPERBOLA. The standard form equation of a hyperbola with center at (h,k) is: 1. When the transverse axis is horizontal: 2. When the transverse axis is vertical: The vertices are a units from the center, and foci are c units from the center, where c2=a2+b2.

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**WRITE THE STANDARD FORM OF THE EQUATION OF A HYPERBOLA**

1. Determine whether the transverse axis is oriented horizontally or vertically. 2. Find the coordinates of the center of the hyperbola. 3. Find the value of c, which is the distance between the center and the foci.

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**4. Find the value of a, which is the distance between**

the center and the vertices. 5. Find the value of b, using the formula 6. Substitute the values of a and b into the standard form of the equation of the hyperbola.

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EXAMPLE 1. Find the standard form of the equation of the hyperbola with foci at (–1,2) and (5,2) and vertices at (0,2) and (4,2).

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**ASYMPTOTES OF A HYPERBOLA**

ASYMPTOTES OF A HYPERBOLA. Each hyperbola has two slant asymptotes that intersect at the center of the hyperbola. 1. When the transverse axis is oriented horizontally, the asymptotes are: 2. When the transverse axis if oriented vertically,

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**EXAMPLE 2. Find the equations of the slant asymptotes of the hyperbola**

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**USING ASYMPTOTES TO FIND THE STANDARD EQUATION OF A HYPERBOLA**

1. Determine the center of the hyperbola by finding the point of intersection of the asymptotes. Use any method to solve the system of linear equations. 2. Determine the values of a and b. Let m1 be the positive slope, then (horizontal) or (vertical). 3. Substitute the values of h, k, a, and b into the standard form of the equation.

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EXAMPLE 3. Determine the standard form of the equation of the hyperbola with vertices (3,–5) and (3,1) and asymptotes y=2x–8 and y=–2x+4.

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**SKETCHING THE GRAPH OF A HYPERBOLA**

1. Determine whether the transverse axis is oriented horizontally or vertically. 2. Determine the values of a, b, and c, if necessary. 3. Determine and plot the center of the hyperbola. 4. Determine and plot the vertices, of the hyperbola. a. If the transverse axis is oriented horizontally, the vertices are (h+a,k) and (h–a,k) .

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**b. If the transverse axis is oriented vertically, the**

vertices are (h,k+a) and (h,k–a). 4. Determine the conjugate axes. a. If the transverse axis is oriented horizontally, the conjugate axes are (h,k+b) and (h,k–b). conjugate axes is (h+b,k) and (h–b,k). 5. Construct a box that passes through the both vertices and the points marking the conjugate axes.

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**6. Determine the equation of the slant asymptotes.**

Then graph the slant asymptotes. The slant asymptotes pass through opposite corners of the box constructed in Step 5. 7. Draw the curve of the hyperbola, which passes through both vertices and converges to the slant asymptotes.

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**EXAMPLE 4. Sketch the hyperbola**

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**GENERAL FORM OF THE EQUATION OF A CONIC SECTION**

GENERAL FORM OF THE EQUATION OF A CONIC SECTION. The general form of the equation of a conic section is:

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**CLASSIFYING CONIC SECTIONS**

CLASSIFYING CONIC SECTIONS. To classify the type of conic section, given the general form of the equation, use the following rules. 1. If A=C, then the conic section is a circle. 2. If AC=0, then the conic section is a parabola. 3. If AC>0, then the conic section is an ellipse. 4. If AC<0, then the conic section is a hyperbola.

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**EXAMPLE 5. Classify each conic section.**

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CLASS WORK/HOMEWORK: SECTION: 10 – 3 PAGE: 720 – 721 PROBLEMS: 7, 9, 11, 23, 25, 27, 29, 39, 44 – 51 All

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