Q2-2-Hyperbola-with-Center-at-the-Origin.pptx

BasCal Hyperbola

1. HYPERBOLA WITH CENTER AT THE ORIGIN

2. MELCs Define a hyperbola. Determine the standard form of equation of a hyperbola.

3. Objectives State the definition of hyperbola. Recognize the parts of the hyperbola given their corresponding description.

4. Objectives Identify the value of the parts of the hyperbola given its equation in standard form. Find the equation of the hyperbola in standard form determined by the given conditions.

5. HYPERBOLA The difference of the distances between the foci and a point on the hyperbola is fixed.

6. HYPERBOLA The set of all points such that the absolute value of the difference of the distance of each point from two fixed points is a constant.

7. PARTS OF THE HYPERBOLA

8. The line passing through the two foci intersects the hyperbola at its vertices .   The line segment that is perpendicular to the conjugate axis and contains the vertices is called the transverse axis . Midpoint of the transverse axis is called the center of the hyperbola The line segment perpendicular to the transverse axis and passing through the center is the conjugate axis .   Asymptotes are straight lines that contain the diagonals of a rectangle drawn at the center of the hyperbola The distance between the vertex and the center is a, while the distance between the focus and the center is c. We get the value of c using the relationship c 2 =a 2 +b 2  

9. Properties of a Hyperbola with Center at (0,0)

10. Horizontal Vertical Equation Center (0,0) (0,0) Vertices (V) (± a , 0) (0, ± a) End of conjugate axis (ECA) (0 ,± b) (± b, 0) Foci ( c ) C (±c , 0) C (0, ±c) Length of the latus Rectum Asymptotes y =± y =± Length of the transverse axis (LTA) 2a 2a Length of the conjugate axis 2b 2b Graph Horizontal Vertical Equation Center (0,0) (0,0) Vertices (V) (± a , 0) (0, ± a) End of conjugate axis (ECA) (0 ,± b) (± b, 0) Foci ( c ) Length of the latus Rectum Asymptotes Length of the transverse axis (LTA) 2a 2a Length of the conjugate axis 2b 2b Graph

11. Equation Transverse Axis Vertices Foci Ends of Conjugate Axis c>a c=   Horizontal (±a,0) (±c,0) c>a c=     (0,±a) (0,±c) Asymptotes     Equation Vertical (0,±b) (±b,0)

12. Example 1: Identify the coordinates of the vertices , foci, ends of conjugate axis, and equation of the asymptotes.  

13. Step 1: Identify the values of a and b from the standard equation. a 2 =16 b 2 =4 a=±4 b=±2  

14. Step 2: Determine the vertices. a=distance between the vertex and the center which is at the origin Horizontal Transverse Axis Vertices at (‒4,0) and (4,0) .

15. Step 3: Determine the ends of the conjugate axis. b=distance from the center to the endpoints of the conjugate axis (0,‒2) and (0,2)

16. Step 4: Use the relationship c 2 =a 2 +b 2 to determine the foci and substitute the values of a and b into the equation. c 2 =a 2 +b 2 c 2 =16+4 c 2 =20 c=± c=± ≈4.5 (4.5,0) and (‒4.5, 0)  

17. Step 5: To determine the equations of the asymptotes, substitute the values of a and b to the equation y=± x. y=± x y=± x y=± x  y= x and y= ‒ x  

18. WRITING THE STANDARD FORM OF THE EQUATION OF A HYPERBOLA Example 1

19. Example 1: Find the standard form of the equation of the hyperbola with center at the origin, foci (±7,0) , and vertices (±5,0).

20. Step 1: Determine the value of a. Since a is the distance between the center (0,0) and the vertex (5,0), a=5 Furthermore, since the vertices lie on the x-axis, the hyperbola has a horizontal transverse axis .

21. Step 2: Determine the value of c. Since c is the distance between the center (0,0) and the focus (7,0), c=7

22. Step 3: Determine the value of b. Using the relationship c 2 =a 2 +b 2 , it follows that b 2 =c 2 ‒a 2 b 2 =7 2 ‒5 2 b 2 =49‒25 b 2 =24 b= b= b=±2 ≈±4.9  

23. Step 4: Determine the standard form of the equation of the hyperbola. Since the hyperbola has a horizontal transverse axis, we use the form ‒ =1. Upon substituting the values, we get ‒ = 1 .  

24. Step 5: Simplify the standard form of the equation of the hyperbola. ‒ = 1   Therefore, the standard form of the equation of the hyperbola is ‒ = 1  

25. WRITING THE STANDARD FORM OF THE EQUATION OF A HYPERBOLA Example 2

26. Example 2: Find the standard form of the equation of the hyperbola having vertices (0, ±10) and asymptotes y=±5x.

27. Step 1: Determine the center of the hyperbola by getting the midpoint between the vertices. Midpoint = =(0,0) Thus, the center of the hyperbola is (0,0)  

28. Step 2: Determine the value of a. Since a is the distance from the center to the vertex, then a=10. Furthermore, since the vertices lie on the y-axis, the hyperbola has a vertical transverse axis.

29. Step 3: Determine the slopes of the asymptotes. Since the hyperbola has a vertical transverse axis, asymptotes are of the form y=± x. thus, y= ±5x. Therefore the slopes are m 1 = 5 and m 2 = ‒5.  

30. Step 4: Determine the value of b. Since 5= , then b= Thus, b= =2  

31. Step 5: Determine the standard form of the equation of the hyperbola. Since the hyperbola has a vertical transverse axis with center (0,0) , we use the standard form of the equation ‒ Substituting the values, we get ‒ or ‒ Therefore, the standard form of the equation of the hyperbola is ‒