Introduction to Conic Sections: Circle Equations and Tangents

1. 10-1 Introduction to Conic Sections Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

2. Warm Up Solve for y. 1. x2 + y2 = 1 2. 4x2 – 9y2 = 1

3. Objectives Recognize conic sections as intersections of planes and cones. Use the distance and midpoint formulas to solve problems.

4. Vocabulary conic section

5. Every conic section can be defined in terms of distances. You can use the Midpoint and Distance Formulas to find the center and radius of a circle.

6. In Chapter 5, you studied the parabola. The parabola is one of a family of curves called conic sections.Conicsections are formed by the intersection of a double right cone and a plane. There are four types of conic sections: circles, ellipses, hyperbolas, and parabolas. Although the parabolas you studied in Chapter 5 are functions, most conic sections are not. This means that you often must use two functions to graph a conic section on a calculator.

7. A circle is the set of points in a plane that are a fixed distance, called the radius, from a fixed point, called the center. Because all of the points on a circle are the same distance from the center of the circle, you can use the Distance Formula to find the equation of a circle.

8. Helpful Hint The midpoint formula uses averages. You can think of xM as the average of the x-values and yM as the average of the y-values. Because a diameter must pass through the center of a circle, the midpoint of a diameter is the center of the circle. The radius of a circle is the distance from the center to any point on the circle and equal to half the diameter.

9. 4–8 2 ( , ) = (2.5, –2) 5 + 0 2 Example 3: Finding the Center and Radius of a Circle Find the center and radius of a circle that has a diameter with endpoints (5, 4) and (0, –8). Step 1 Find the center of the circle. Use the Midpoint Formula with the endpoints (5, 4) and (0, –8).

10. r = ( 5 – 2.5)2 + (4 – (–2))2 Example 3 Continued Step 2 Find the radius. Use the Distance Formula with (2.5, –2) and (0, –8) The radius of the circle is 6.5 CheckUse the other endpoint (5, 4) and the center (2.5, –2). The radius should equal 6.5 for any point on the circle.  The radius is the same using (5, 4).

11. 6+ 22 2 ( , ) = (8, 14) 2 + 14 2 Check It Out! Example 3 Find the center and radius of a circle that has a diameter with endpoints (2, 6) and (14, 22). Step 1 Find the center of the circle. Use the Midpoint Formula with the endpoints (2, 6) and (14, 22).

12. r = ( 2 – 8)2 + (6 – 14)2 Check It Out! Example 3 Continued Step 2 Find the radius. Use the Distance Formula with (8, 14) and (14, 22). The radius of the circle is 10. Check Use the other endpoint (2, 6) and the center (8, 14). The radius should equal 10 for any point on the circle.  The radius is the same using (2, 6).

13. Notice that r2 and the center are visible in the equation of a circle. This leads to a general formula for a circle with center (h, k) and radius r.

14. Example 2B: Writing the Equation of a Circle Write the equation of the circle. the circle with center (–4, 11) and containing the point (5, –1) Use the Distance Formula to find the radius. (x + 4)2 + (y – 11)2 = 152 Substitute the values into the equation of a circle. (x + 4)2 + (y – 11)2 = 225

15. Check It Out! Example 2 Find the equation of the circle with center (–3, 5) and containing the point (9, 10). Use the Distance Formula to find the radius. Substitute the values into the equation of a circle. (x + 3)2 + (y – 5)2 = 132 (x + 3)2 + (y – 5)2 = 169

16. A tangent is a line in the same plane as the circle that intersects the circle at exactly one point. Recall from geometry that a tangent to a circle is perpendicular to the radius at the point of tangency.

17. From the equation x2 + y2 = 29, the circle has center of (0, 0) and radius r = . Example 4: Writing the Equation of a Tangent Write the equation of the line tangent to the circle x2 + y2 = 29 at the point (2, 5). Step 1 Identify the center and radius of the circle.

18. 2 – 5 5 2 The slope of the radius is . Because the slopes of perpendicular lines are negative reciprocals, the slope of the tangent is . Example 4 Continued Step 2 Find the slope of the radius at the point of tangency and the slope of the tangent. Use the slope formula. Substitute (2, 5) for (x2 , y2 ) and (0, 0) for (x1 , y1 ).

19. Step 3 Find the slope-intercept equation of the tangent by using the point (2, 5) and the slope m = . 2 – 5 Substitute (2, 5) (x1 , y1 ) and – for m. 2 5 Example 4 Continued Use the point-slope formula. Rewrite in slope-intercept form.

20. The equation of the line that is tangent to x2 + y2 = 29 at (2, 5) is . Example 4 Continued Check Graph the circle and the line.

21. Check It Out! Example 4 Write the equation of the line that is tangent to the circle 25 = (x –1)2 + (y + 2)2, at the point (1, –2). Step 1 Identify the center and radius of the circle. From the equation 25 = (x – 1)2 +(y + 2)2, the circle has center of (1, –2) and radius r = 5.

22. –3 4 The slope of the radius is . Because the slopes of perpendicular lines are negative reciprocals, the slope of the tangent is . Check It Out! Example 4 Continued Step 2 Find the slope of the radius at the point of tangency and the slope of the tangent. Use the slope formula. Substitute (5, –5) for (x2 , y2 ) and (1, –2) for (x1 , y1 ).

23. Step 3. Find the slope-intercept equation of the tangent by using the point (5, –5) and the slope . Substitute (5, –5 ) for (x1 , y1 ) and for m. 4 3 Check It Out! Example 4 Continued Use the point-slope formula. Rewrite in slope-intercept form.

24. Check It Out! Example 4 Continued The equation of the line that is tangent to 25 = (x – 1)2 + (y + 2)2 at (5, –5) is . Check Graph the circle and the line.