Equations of Circles.

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Presentation transcript:

Equations of Circles

Centre-radius form of a circle Center is at (h, k) r is the radius of the circle (h,k) r

General Form of a Circle Expand the brackets and make the RHS equal to 0. Every term is on the left side, equal to 0.

EX 1 Write an equation of a circle with center (3, -2) and a radius of 4. k r

EX 2 Write an equation of a circle with center (-4, 0) and a diameter of 10. k

EX 3 Write an equation of a circle with center (2, -9) and a radius of . k r

Opposite signs! ( , ) 6 -3 Take the square root! Radius 5 EX 4 Find the coordinates of the center and the measure of the radius. Opposite signs! ( , ) 6 -3 Take the square root! Radius 5

5. Find the center, radius, & equation of the circle. The center is The radius is The equation is (0, 0) 12 x2 + y2 = 144

6. Find the center, radius, & equation of the circle. (1, -3) The center is The radius is The equation is 7 (x – 1)2 + (y + 3)2 = 49

7. Graph the circle, identify the center & radius. (x – 3)2 + (y – 2)2 = 9 Center (3, 2) Radius of 3

Converting from general form to centre-radius form Move the x terms together and the y terms together. Move C to the other side. Complete the square (as needed) for x. Complete the square(as needed) for y. Factor the left & simplify the right.

8. Write the centre-radius equation of the circle 8. Write the centre-radius equation of the circle. State the center & radius. Center: (4, 0) radius: 3

Using TI Nspire CAS In the calculator screen, select Menu, Algebra, Complete the square, Type completeSquare(x2+y2-8x=7,x,y) The calculator will return (x-4)2+y2=23 Equate the above expression to 0 and move 9 to the right-hand side. Refer to Ti Nspire CAS handout how to sketch circles on your calculator.

9. Write the centre-radius equation of the circle 9. Write the centre-radius equation of the circle. State the center & radius. Center: (-2, 3) radius: 4

10. Write the centre-radius equation of the circle 10. Write the centre-radius equation of the circle. State the center & radius.

11. Write the general form of the equation of the circle given in centre-radius form.

Semicircles Given the equation of a circle, we make y the subject. Example: Write down the equation of the upper and lower semicircle, given Therefore the equation of the upper semicircle is And the equation of the lower semicircle is

Shading regions Draw a circle and test with a point whether it is inside or outside. In general is the region outside the circle without the boundary so a dotted line. And is the region inside the circle, including the boundary; thus a continuous line.

Homework Exercise 4D Q1 d, e Q2 a, d Q3 c, f Q4 b, d by hand; Q4 a, e on the calculator Q5 b Q6 b, d Q7a Exercise 4E Q 9 Q 14 a