1. Conic Sections:Circles
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2. Introduction of Conics
A conic (or conic section) is a plane curve that can be obtained by intersecting a cone
Four shapes that make up conic sections: circle, ellipse, hyperbola, and parabola
9.3– Conic Sections: Circles
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3. Shape of a circle from a cone
9.3– Conic Sections: Circles
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4. 9.3– Conic Sections: Circles
Real–Life Examples
9.3– Conic Sections: Circles
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5. 9.3– Conic Sections: Circles
Definitions
A circle is the set of all points in a plane that is a fixed distance from a fixed point
The center of a circle is the fixed equidistance point of the circle
The radius of a circle is the fixed equidistance length of the circle
A tangent is a line in the same plane as the circle that intersects at exactly one point
9.3– Conic Sections: Circles
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6. Equations in Standard Form
Center: (h, k)
Radius: r
9.3– Conic Sections: Circles
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7. 9.3– Conic Sections: Circles
Example 1
Name the center and the radius of this circle, (x + 2)2 + (y – 3)2 = 16
9.3– Conic Sections: Circles
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8. 9.3– Conic Sections: Circles
Your Turn
Name the center and the radius of this circle, (x + 1/2)2 + (y – 5/2)2 = 14
9.3– Conic Sections: Circles
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9. 9.3– Conic Sections: Circles
Example 2
Identify the equation of a circle with the given graph below.
Each line is 0.5
9.3– Conic Sections: Circles
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10. 9.3– Conic Sections: Circles
Example 3
Write an equation of the circle with its center is at (–2, 3), the radius is 4, and graph it.
9.3– Conic Sections: Circles
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11. Example 3 Calculator Check
Write an equation of the circle with its center is at (–2, 3), the radius is 4, and graph it.
9.3– Conic Sections: Circles
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12. 9.3– Conic Sections: Circles
Your Turn
Write an equation of the circle with its center is at (–3, 5) and the radius is 5 and graph it.
9.3– Conic Sections: Circles
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13. 9.3– Conic Sections: Circles
Example 4
Write and graph the equation in standard form with the given center of (2, –3), the point (1, 0), and identify the radius.
Plug in what you are given to figure out the radius
9.3– Conic Sections: Circles
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14. 9.3– Conic Sections: Circles
Example 5
Write a circle equation whose center is at the origin and passes through (1, –6).
9.3– Conic Sections: Circles
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15. 9.3– Conic Sections: Circles
Your Turn
Write a circle equation whose endpoints of a diameter are (–5, 2) and (3, 6).
9.3– Conic Sections: Circles
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16. 9.3– Conic Sections: Circles
Tangent Steps
Determine the two points – usually the center and a point
Determine the slope of radius and take the OPPOSITE SIGN RECIPROCAL
Use point–slope formula to write equation
9.3– Conic Sections: Circles
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17. 9.3– Conic Sections: Circles
Example 6
Write the equation of the line tangent to the circle x2 + y2 = 29 at the point (2, 5).
Put the equation in slope–intercept form using POINT–slope form with the center and point
The perpendicular lines are negative reciprocals, the slope of the tangent is –2/5.
9.3– Conic Sections: Circles
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18. 9.3– Conic Sections: Circles
Example 6
Write the equation of the line tangent to the circle x2 + y2 = 29 at the point (2, 5).
Put the equation in slope–intercept form using POINT–slope form with the center and point
You can leave it in POINT–SLOPE form
9.3– Conic Sections: Circles
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19. 9.3– Conic Sections: Circles
Example 6
Write the equation of the line tangent to the circle x2 + y2 = 29 at the point (2, 5).
9.3– Conic Sections: Circles
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20. 9.3– Conic Sections: Circles
Example 7
Write the equation of the line tangent to the circle (x – 1)2 + (y + 2)2 = 25 at the point (5, –5).
9.3– Conic Sections: Circles
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21. 9.3– Conic Sections: Circles
Your Turn
Write the equation of the line tangent to the circle (x – 1)2 + (y + 2)2 = 26 at the point on the circle (2, 3).
9.3– Conic Sections: Circles
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22. 9.3– Conic Sections: Circles
Assignment
Worksheet
9.3– Conic Sections: Circles
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