CIRCLES page 156 LINGKARAN halaman 156 Standar Kompetensi
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CIRCLES page 156 LINGKARAN halaman 156
Standar Kompetensi Menyusun persamaan lingkaran dan garis singgungnya.
Kompetensi Dasar 3. 1. Menyusun pers. lingkaran yg memenuhi persyaratan yg ditentukan - Merumuskan pers. lingkaran berpusat di (0, 0) dan (a, b). - Menentukan pusat dan jari-jari lingkaran yg pers. nya diketahui. - Menentukan pers. lingkaran yg memenuhi kriteria tertentu. 3. 2. Menentukan pers. garis singgung pd lingkaran dalam berbagai situasi - Melukis garis yg menyinggung lingkaran dan menentukan sifatnya - Merumuskan pers. garis singgung yg melalui suatu titik pd lingk. - Merumuskan pers. garis singgung yg gradiennya diketahui.
3. 1. Menyusun pers. lingkaran yg memenuhi persyaratan yg ditentukan - Merumuskan pers. lingkaran berpusat di (0, 0) dan (a, b). - Menentukan pusat dan jari-jari lingkaran yg pers. nya diketahui. - Menentukan pers. lingkaran yg memenuhi kriteria tertentu.
Jadwal Ulangan Lingkaran 1 Lingkaran 2 11 IPA 1 Kamis, 18 Nov Kamis, 2 Des 11 IPA 2 Selasa, 16 Nov Selasa, 30 Nov 11 IPA 3 Jumat, 19 Nov Rabu, 1 Des
Just what you need Distance between 2 points Gradient (m) B(x. B, y. B) y A (x. A, y. A) x
Just what you need ñ Linear (line) equation Distance between point & line y → M(p, q) a r ax + by = ab Change it into: ax + by + c = 0 b x
Exp. 1 Exp. 2 Exp. 3 Calculate the distance between (2, 7) & (– 1, 3) and its line gradient. Find the line equation through point (6, 0) & (0, 8) Find the distance (r) between point (2, – 5) and line 4 x – 3 y = 12 Answer: x = 2 – (– 1) = 3 Change 4 x – 3 y = 12 into 4 x – 3 y – 12 = 0 8 y = 7 – 3 = 4 Point (2, – 5) 6 8 x + 6 y = 48 4 x + 3 y = 24
CIRCLE EQUATION Circle equation with center (0, 0) is : x 2 + y 2 = r 2 r y x Exp. 4 Find circle equation which its center is (0, 0) and has radius of 3 units. Answer: The equation is: x 2 + y 2 = 9 r
Position of a point to the circle S Point J lies inside the circle so we said that: m 2 + n 2 < r 2 r y x J (m, n) Point S is on the circle F Point F is outside the circle
Do Exercises no: 1 b 2 b 4 b 6 a page 161
CIRCLE EQUATION Circle equation with center P(a, b) is : (x–a)2 + (y–b)2 = r 2 r b P Exp. 5 Find circle equation which its center is (2, 3) and has radius of 5 units. Answer: a The equation is: (x– 2)2 + (y– 3)2 = 52 or x 2 + y 2 – 4 x – 6 y – 12 = 0
Exp. 6 Exp. 7 Find the circle equation which its center lies on P(2, 3) and the circle passes through point (– 1, 4) A circle has diameter line that passes through (– 5, 1) and (3, – 7). Find its equation. Answer: (– 5, 1) (– 1, 4) P P First, calculate the radius (3, – 7) First, find the coordinate of P Second, find the radius The equation: (x – 2)2 + (y – 3)2 = 10 x 2 + y 2 – 4 x – 6 y + 3 = 0 (x – (– 1))2 + (y – (– 3))2 = 32 x 2 + y 2 + 2 x + 6 y – 22 = 0
ANOTHER FORM OF CIRCLE EQUATION In the last equation: (x–a)2 + (y–b)2 = r 2 with center P(a, b) and radius r another form is: x 2 + y 2 + Ax + By + C = 0 with center and radius the coefficient of x 2 and y 2 must be 1
Exp. 8 Exp. 9 Find the center of circle: x 2 + y 2 + 4 x – 6 y + 10 = 0 A circle of x 2 + y 2 – 4 x + 2 y + C = 0 passes through (5, – 1). Find its radius. Answer: A=4 B = – 6 First, find the value of C then insert (5, – 1) into the equation Center: P(–A/2, –B/2) 52 + (– 1)2 – 4. 5 + 2 (– 1) + C = 0 C = – 4 P(– 4/2, –(– 6)/2) = (– 2, 3)
Exp. 10 Exp. 11 Find the center and radius of: 2 x 2 + 2 y 2 + 16 x – 12 y + 6 = 0 Circle x 2 + y 2 – 4 x + 6 y + k = 0 has a radius of 5 units. Find the value of k. Answer: First, divide it by 2 x 2 + y 2 + 8 x – 6 y + 3 = 0 Answer:
Do Exercises page 164 no: 1 b 2 b 3 5 b 6 b 7 1 b 2 b 3 b 5 8 9 page 167 no: Daily test 8: Circles 1 4 b
Next KD. . . 3. 2. Menentukan pers. garis singgung pd lingkaran dlm berbagai situasi - Melukis garis yg menyinggung lingkaran & menentukan sifatnya - Merumuskan pers. garis singgung yg melalui suatu titik pd lingk. - Merumuskan pers. garis singgung yg gradiennya diketahui.
TANGENT LINE ON A CIRCLE Tangent (line) is a line that touch the circle in one point only. Line touches the circle at point E, then line is a tangent of the circle. E r Line also a tangent of the circle. F r Line is not a tangent of the circle, because it intersects the circle at two points. P G H
PROPERTIES OF A TANGENT A circle has so many tangents depend on the position of the point. D K If the point lies inside the circle (point K), then there is no tangent, because the line must be intersects the circle at 2 points. If the point lies on the circle exactly (point T), then there is 1 tangent only. P T If the point lies outside the circle (point D), then there are 2 tangents. How do we know the position of the point to the circle? just insert the point coordinate into the circle equation.
FINDING THE TANGENTS EQUATION Case 1: Point on the circle STEPS: » Find the circle center r P (a, b) T (x 1, y 1) » Find the gradient of line PT (m. PT) » Because of line PT is perpendicular to line then: m. PT. m 1 = – 1 » Use line equation formula to find the tangent equation: y – y 1 = m 1 (x – x 1)
Exp. 12 Exp. 13 Find the tangent line on circle (x – 2)2 + (y + 1)2 = 16 that passes through H(5, – 1) Find the tangent line on circle x 2 + y 2 – 8 x + 6 y – 20 = 0 that passes through J(1, – 9) Answer: Check the position of point J(1, – 9) 12 + (– 9)2 – 8. 1 + 6 (– 9) – 20 = ? 1 + 81 – 8 – 54 – 20 = 0 P (2, – 1) The center: From the graph above, let’s check the position of point H(5, – 1) (x – 2)2 + (y + 1)2 = 16 H(5, – 1) (5 – 2)2 + (– 1 + 1)2 = 9 < 16 There is no tangent line. RALAT Gradient PJ: Gradient of tangent: m. PJ. m 1 = – 1/2 Tangent equation: y – y 1 = m 1 (x – x 1) y – (– 9) = – 1/2 (x – 1) 2 y + x + 17 = 0
Exp. 13 Exp. 14 Find the tangent line on circle x 2 + y 2 – 8 x + 6 y – 20 = 0 that passes through J(1, – 9) Another way to find the tangent on circle x 2 + y 2 – 8 x + 6 y – 20 = 0 that passes through J(1, – 9) Answer: Check the position of point J(1, – 9) Change x 2 + y 2 – 8 x + 6 y – 20 = 0 12 + (– 9)2 – 8. 1 + 6 (– 9) – 20 = ? into form (x – a)2 + (y – b)2 = r 2 1 + 81 – 8 – 54 – 20 = 0 x 2 – 8 x + 16 + y 2 + 6 y + 9 = 20 + 16 + 9 The center: (x – 4)2 + (y + 3)2 = 45 Gradient PJ: Change it: (x 1– 4)(x– 4) + (y 1+3)(y+3) = 45 Gradient of tangent: Insert J(1, – 9) into the equation m. PJ. m 1 = – 1/2 Tangent equation: y – y 1 = m 1 (x – x 1) y – (– 9) = – 1/2 (x – 1) 2 y + x + 17 = 0 (1– 4)(x– 4) + (– 9 +3)(y+3) = 45 – 3(x– 4) – 6(y+3) = 45 – 3 x + 12 – 6 y – 18 = 45 2 y + x + 17 = 0
Exp. 15 Exp. 16 Find the tangent line on circle x 2 + y 2 = 65 that passes through G(8, 1) Find the tangent line on circle x 2 + y 2 – 8 y = 37 that passes through Z(2, – 3) Answer: Check the position of point G(8, 1) Check the position of point Z(2, – 3) 82 + 12 = 65 (on the circle) 22 + (– 3)2 – 8(– 3) = ? x 2 + y 2 = 65 x. x + y. y = 65 4 + 9 + 24 = 37 (on the circle) x 2 + y 2 – 8 y = 37 x 1. x + (y– 4)2 = 53 insert G(8, 1) insert Z(2, – 3) 8 x + 1 y = 65 8 x + y = 65 2 x + – 7(y– 4) = 53 2 – 7 y = 25 RALAT
Do Exercises page 175 no: 1 b 5 2 b 3 7 9 4
FINDING THE TANGENTS EQUATION Case 2: the gradient is known STEPS: r T (x 1, y 1) » Find the circle center P(a, b) » Find the radius (r) P (a, b) : » The tangent equation is: because there will be 2 tangent lines (one parallel to another)
Exp. 17 Exp. 18 A circle has equation: (x – 5)2 + y 2 = 49 Find its tangent equation if the gradient is 3. If line y = x + k touches circle x 2 + y 2 = 25 then find the value of k. Answer: x 2 + y 2 = 25 P(0, 0) and r = 5 3 P(5, 0) 1 y =x+k m=1 y – 0 = 1 (x – 0) + k k 2 = r 2 (1 + m 2) = 52 (1 + 12) = 50
FINDING THE TANGENTS EQUATION Case 3: the point is outside the circle STEPS T(x 1, y 1) » Find the polar line equation by changing x 2 + y 2 = r 2 aaaainto x 1 x + y 1 y = r 2 » Insert the polar equation into circle equation to find x 1 and x 2 P (a, b) » Find y 1 and y 2 and the two coordinates (x 1, y 1) and (x 2, y 2) » Insert each coordinate into circle equation x 1 x + y 1 y = r 2 to find the tangents equation.
Exp. 19 Exp. 20 Find the tangents line to the circle x 2 + y 2 = 25 that passes through (7, 1) Find the tangents line to the circle x 2 + y 2 – 2 x – 6 y – 15 = 0 that passes through point (6, – 2) Answer: (7, 1) lies outside the circle (6, – 2) lies outside the circle Polar: x 1 x + y 1 y = 25 7 x + y = 25 – 7 x x 2 + y 2 = 25 x 2 + (25 – 7 x)2 = 25 5(x – 1) – 5(y – 3) = 35 5 x – 5 y + 10 = 35 y = x – 5 x 2 – 7 x + 12 = 0 (x – 3)(x – 4) = 0 x = 3 y = 4 (3, 4) x = 4 y = – 3 (4, – 3) (3, 4) 3 x + 4 y = 25 (4, – 3) 4 x – 3 y = 25 Polar: (x – 1)2 + (y – 3)2 = 35 (6 – 1)(x – 1) + (– 2 – 3)(y – 3) = 35 (x – 1)2 + (x – 5 – 3)2 = 35 x 2 – 9 x + 15 = 0 use abc formula RALAT Do Exercises page 180 no: 1 c 2 c 5
Do Exercises page 183 and from buku Mandiri page 56 – 70 FINISH
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