Chapter 12: Quadratic Relations and Analytic Geometry

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Chapter 12: Quadratic Relations and Analytic Geometry

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1. Find the distance between the two points.

(-5, 8), (0, -4)

2. Find the distance between the two points.

(-3 √ 7, 6), (3 √ 7, 4)

3. Solve for x using the given distance d between the two points.

4. The coordinates of one endpoint of the line segment XY and the midpoint M is given. Find the coordinates of the other end point.

M(0.55, 2.95), X(4.2, 7.8)

5. Classify the triangle as scalene, isosceles, or equilateral from the given vertices.

(7, 0), (3, -4), (8, -5)

6. Write an equation and graph a circle with center (-1, 1) and radius 6.

7. Find the equation of the circle with center (1.5, 0.3) and r = 11.5 units.

8. Consider the circle (x + 5)² + (y - 7)² = 41.

Find its center and radius.

9. Graph the circle (x - 5)² + y² = 25.

10. Write an equation of the line that is tangent to the circle x² + y² = 85 at the point (-6, -7).

11. The coordinates of vertex of a parabola are V(3, 2) and equation of directrix is y = -2. Find the coordinates of focus.

12. The coordinates of focus and vertex of a parabola are F(2, -2) and V(2, -5) respectively. Find the equation of directrix.

13. Without graphing, decide whether the parabola opens up, down, left, or right.

9y² = x

14. Find the focus and directrix of the parabola.

x² = -24y

15. Find an equation of parabola with focus (2, 4) and vertex (2, 2), and graph it.

16. Find an equation of that ellipse that has x-intercepts: ± 4,y-intercepts: ± 11, and center at the origin.

17. Graph the equation and then find the vertices, co-vertices, and foci of the ellipse.

18. Find the points of intersection of the following graphs:

y = x + 2

19. Rewrite the equation of the hyperbola in standard form.

y² - 9x² = 36

20. Rewrite the equation in standard form. Identify the vertices and foci of the hyperbola.

49y² - 25x² = 1225

21. The circle x² + y² = 100 after a translation becomes the circle

(x - 15)² + (y + 8)² = 100. Find the translation.

22. Identify the conic and find its characteristics:

23. Write an equation and graph the conic section:

Hyperbola with center at (1, 2), one focus at (1, 2 + √ 45), one vertex at (1, 5).

24. Tell which conic (circle, parabola, hyperbola, or ellipse) is defined by the equation

x² + 6x - y + 5 = 0.

25. Tell which conic (circle, parabola, hyperbola, or ellipse) is defined by the equation

y² - 3x² + 3x + 2y = 14.

26. Tell which conic (circle, parabola, hyperbola, or ellipse) is defined by the equation

3x² + 3y² - 7x - y - 6 = 0.

27. Find the number of real solutions of the system

9x² + 25y² = 225 and 2x + 3y = 3 by sketching graphs.

28. What are the point(s) of intersection of the line y = x - 2 and the parabola y = 4x²?

29. Find the real solutions of the system

2x² - 3y²= 18

x² + y² = 19

30. Find the points, if any, that the graphs of all three equations have in common.

x² + y² - 5x - 3y = 46

x² + y² - 5x = 70

y = 2x - 4

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