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P arabolas Circles Ellipses

Presented by: Ihda Mardiana H. Hesti Setyoningsih Dewi Kurni y ati Belynda Surya F. P arabolas Circles Ellipses. PARABOLAS. Bentuk Umum Persamaan kuadrat D engan A, B, C, D, E, dan F adalah bilangan real. Jika B,C = 0, maka persamaanya menjadi: Atau biasa di tulis dalam bentuk:

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P arabolas Circles Ellipses

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  1. Presented by: • Ihda Mardiana H. • Hesti Setyoningsih • Dewi Kurniyati • Belynda Surya F. ParabolasCirclesEllipses

  2. PARABOLAS

  3. Bentuk Umum Persamaan kuadrat Dengan A, B, C, D, E, dan F adalah bilangan real. • Jika B,C = 0, maka persamaanya menjadi: Atau biasa di tulis dalam bentuk: • Jika A,B = 0, maka persamaanya menjadi: Atau biasa di tulis

  4. Definition 2.1 Parabola adalahtempatkedudukan (locus) titik-titikpadasebuahbidangdatar yang jaraknyaterhadapsebuahgaris yang tetap (a fixed line) yang disebutdirektrik, danterhadaptitik yang tetap (a fixed point) yang disebutfocus adalahsama. A parabola is the locus of points in a plane whose distance froma fixed (the dirictrix), and a fixed point (the focus) are equal. d1 parabola focus d2 vertex directrix d1=d2 Axis Line of symmetry

  5. Standard Equation ( Persamaan Baku ) Suatu parabola memilikipersamaanbakujikadanhanyajikakoordinatfokusnya (0,p/2) dandirektriknyamemilikipersamaan A parabola has the standard equation if and only if its focus has the coordinates (0,p/2) and its directrix has an equation Y Teorema2.1 d1 d2

  6. Bukti Teorema 2.1 Berdasarkan definisi parabola, Y d1 d2

  7. Theorem 2.2 Parabola memilikipersamaanbaku Jikadanhanyajikafokusnyamemilikikoordinat (0, Dan direktriknyamemilikipersamaan A parabola has the standard equation If and only if its focus has the coordinates (0, Its directrix has an equation

  8. Bukti Teorema 2.2 Berdasarkandefinisi parabola

  9. Theorem 2.3 Suatu parabola memilikipersamaanbakujikadanhanyajikafokusnyamemilikikoordinat (,0) dandirektriknyamemilikipersamaan A parabola has the standard equation if and only if its focus has the coordinates (,0) and its directrix has an equation

  10. Fokus dan persamaan direktrik

  11. Teorema 2.3 A parabola has the standard equation if and only if its focus has the coordinates (-p/2,0) and its directrix has an equation Parabola memilikipersamaanbakujikadanhanyajikafokusnyamemilikikoordinat(-p/2,0) dandirektriknyamemilikipersamaan

  12. Fokus dan persamaan direktrik

  13. Test Of Symmetry If is replaced with in the equation and we have, and Thus, the equation and the resulting value of are unchange after the subtitution. Jikadigantikandengan dalam persamaan dan kita mempunyai, dan Persamaan dan nilai hasil dari y tidak berubah setelahdisubstitusikan.

  14. Test Of Symmetry If is replaced with in the equation and we have, and Thus, the equation and the resulting value of are unchange after the subtitution. Jikadigantikandengan dalam persamaandan kita mempunyai, dan Persamaan dan nilai hasil dari y tidak berubah setelahdisubstitusikan. y (x,y) x (x,-y)

  15. Example Find an equation of the directrix and the coordinates the focus of the parabola whose equation is . Sketch the curve. Solution: • From theorem 2.3 the given equation is in standard form. 2p=12 hence p/2=3. the directrix has an equation and the focus has coordinates (3,0). • To sketch the curve, we subtitute selected values for x, say 1and 3 in the original equation to obtain and • Using we graph (1,3.5), (1,-3.5), (3,6) and (3,-6) and the sketch of the parabola using the fact that by theorem 2.3 the graph is symmetric with respect to the axis.

  16. CIRCLES (LINGKARAN)

  17. Definition 2.1 A circle is the set of points in a plane that are at a given distance (the radius) from a fixed point (the center). Lingkaranmerupakanhimpunantitik-titikdalambidangdatar yangberjaraksama (radius)darititikditetapkan (pusat)

  18. Standard Equation of Circles Lingkaran yang berpusat di titik asal (0,0) memiliki persamaan sederhana. Sebagaimana ditunjukkan dalam gambar 2.8, lingkaran dengan titk P (x,y) dan berada pada jarak r dari titik pusat dan memenuhi rumus jarak : Circles that have their center at the origin have simple equation as indicated in figure 2.8, any point on such a circle in at a distance from the origin (0,0), and by the distance formul P P=(x,y) r O

  19. Teorema 2.4 A circle having the origin as its centerand radius have the equation Suatulingkaran yang memiliki titik asal sebagai pusat dan panjang jari-jari 𝑟 memiliki persamaan

  20. Y (2,3) 3 3 Contoh: Lingkaran yang berpusat di titik (0,0) dan memiliki titik (2,3) pada jarak r, didefinisikan sebagai sehinggadiperoleh dan persamaan lingkarannya yaitu

  21. ELLIPSES

  22. Definisi An ellipses is the set of points in a plane such that for each point the sum of its distances from two fixed point (the foci) is constant. Ellipsmerupakanhimpunantitik-titikdalambidangdatarsehinggajumlahjarakuntuksetiaptitiknyadariduatitikyang tetap (foci) adalahtetap. P d1 d2 F1 F2

  23. Pusat ellips merupakantitikpertengahan diantara foci. • Garis yang melalui fokus memotong ellips padadua titik yang disebutvertex • Ruas garis yang menghubungkan dua vertexdisebutsumbu mayor • Ruas garis yang dimuatolehgarisyang tegak lurus dengan sumbu mayor pada pusat yang memotong dua titik pada elips disebutsumbu minor. Minor axis Fokus Fokus vertex vertex center Major axis

  24. Standart Equations of an Ellipses Ellips mempunyaipersamaanbaku Jikadanhanyajikapusatnya di titikasaldanfocinya di sumbux An ellipse has standart equation If and only if its center is the origin and its foci are in the x axis Theorem 2.5

  25. Theorem 2.6 An ellipse has standart equation If and only if its center is the origin and its foci are in the y axis Ellips mempunyaipersamaanbaku Jikadanhanyajikapusatnya di titikasaldanfocinya di sumbu y

  26. Eccentricity of an Ellipse(Eksentrisitasellips) Suatuukuranpemanjangansuatuellipsdiberikanolehperbandingan c/a, yang disebuteksentrisitasellips, dinyatakandenganhuruf e. Saat nilai emendekati 1 (c mendekati a), elips menjadi lebih memanjang. Ketika nilai e mendekati 0 (c mendekati 0), elips mendekati bentuk lingkaran F1 F1 F2 F2 E = 0,7 E = 0,4 F1 F2 E = 0,9

  27. Systems of Equations(Sistem Persamaan) System of two quadratic equations in two variables has at most four real solutions(ordered pairs of real number). However, such a system may have two real and two imaginary solutions, or no real and four imaginary solutions. Generally, these different possibilities can be determined by graphical means. Sistem persamaan kuadrat denganduavariabelmemiliki paling banyakempatsolusi real (pasangan terurut dari bilangan real). Namun, sistem seperti ini mungkin memiliki dua solusi real dan dua solusi imajiner, atau tidak ada solusi real dan empat solusi imajiner. Umumnya perbedaan kemungkinan ini dapat ditentukan dengan cara grafis.

  28. Four Real, distinc intersection two Real, distinc intersection, And two coincident intersection two Real, distinc intersection, And two imajinary intersection All four intersection imajinary

  29. SEKIAN TERIMA KASIH

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