/Tag: Extension 2
1 01, 2019

Ellipses-Tangents & Normals

By |2019-01-01T10:42:43+10:00January 1st, 2019|Tags: , , , |0 Comments

Ellipses-Tangents & Normals Go to the Forum to see more questions or ask your own Formulas used in Ellipses-Tangents & Normals Tangent to Ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ $$\frac{x\,x_1}{a^2}+\frac{y\,y_1}{b^2}=1$$ Normal to Ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ $$\frac{a^2 x}{x_1}+\frac{b^2 y}{y_1}=a^2-b^2$$ Worked examples       Conics [...]

11 03, 2017

Inequalities

By |2017-03-13T14:20:04+10:00March 11th, 2017|Tags: , |0 Comments

Inequalities Cauchy's Theorem $$\frac{a_1+a_2+a_3+...+a_n}{n}=\sqrt[3]{{{a_1}{a_2}{a_3}...{a_n}}}$$   further examples = 10x + 15 - 7x + 14 = 3x + 29 harder examples u = 1 + √x = 1 + x1/2 du/dx = ½x–1/2 du = $$\frac{1}{{2\sqrt x }}dx$$ x = 49 [...]

11 02, 2017

Complex Proofs

By |2017-02-11T11:51:35+10:00February 11th, 2017|Tags: , , |0 Comments

Complex Proofs $$z\in C$$ such that $$\frac{z}{z-i}$$ is real. Show that z is imaginary. Let $$\frac{z}{z-i}=a$$, where a is real. Then $$z=a(z-i)$$ $$z=az-ai$$ $$ai=az-z$$ $$ai=z(a-1)$$ so $$z=\frac{ai}{a-1}$$ $$z=\frac{a}{a-1}i$$ Since a is real, then $$\frac{a}{a-1}$$ is real, and hence $$\frac{a}{a-1}i$$ is imaginary thus z is imaginary     [...]

9 02, 2017

Parabolas

By |2017-02-09T10:58:01+10:00February 9th, 2017|Tags: , , |0 Comments

Hyperbolas   $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ eccentricity: $$e=\sqrt{1+\frac{b^2}{a^2}}$$ PS = e.PM Foci: $$(\pm ae, 0)$$ Directrix: $$x=\pm\frac{a}{e}$$ Asymptotes: $$y=\pm\frac{b}{a}x$$   $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ when a < b eccentricity: $$e=\sqrt{1-\frac{a^2}{b^2}}$$ PS = e.PM Foci: $$(0,\pm ae)$$ Directrix: $$y=\pm\frac{b}{e}$$     worked examples further examples = 10x + 15 - [...]

9 02, 2017

Hyperbolas

By |2019-01-08T11:28:22+10:00February 9th, 2017|Tags: , , |0 Comments

Hyperbolas Go to the Forum to see more questions or ask your own Formulas used in Hyperbolas $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ eccentricity: $$e=\sqrt{1+\frac{b^2}{a^2}}$$ $$b^2=a^2(e^2-1)$$ PS = e.PM Foci: $$(\pm ae, 0)$$ Directrix: $$x=\pm\frac{a}{e}$$ Asymptotes: $$y=\pm\frac{b}{a}x$$ $$|PS-PS'|=2a$$ $$\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$$ eccentricity: $$e=\sqrt{1+\frac{a^2}{b^2}}$$ $$a^2=b^2(e^2-1)$$ PS = e.PM Foci: $$(0, \pm be)$$ Directrix: $$y=\pm\frac{b}{e}$$ Asymptotes: $$y=\pm\frac{b}{a}x$$ [...]

9 02, 2017

Ellipses

By |2018-12-29T12:03:36+10:00February 9th, 2017|Tags: , , |0 Comments

Ellipses Go to the Forum to see more questions or ask your own Formulas used in Ellipses $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ when b < a eccentricity: $$e=\sqrt{1-\frac{b^2}{a^2}}$$ $$b^2=a^2(1-e^2)$$ PS = e.PM Foci: $$(\pm ae, 0)$$ Directrix: $$x=\pm\frac{a}{e}$$ $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ when a < b eccentricity: $$e=\sqrt{1-\frac{a^2}{b^2}}$$ $$a^2=b^2(1-e^2)$$ PS = e.PM Foci: $$(0,\pm ae)$$ Directrix: $$y=\pm\frac{b}{e}$$   [...]

7 02, 2017

Graphs

By |2017-02-07T20:20:59+10:00February 7th, 2017|Tags: , |0 Comments

Graphs Go to the Graphs forum to see more questions or ask your own Topics Covered in Graphs Motion in One Dimension   Formulas Used in Graphs     Graphs Forum Rules of [...]

7 02, 2017

Conics

By |2019-01-08T11:25:45+10:00February 7th, 2017|Tags: , |0 Comments

Conics Go to the Conics forum to see more questions or ask your own Topics Covered in Conics Ellipses Hyperbolas Ellipses-Tangents & Normals Formulas Used in Conics  Ellipse  $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ eccentricity: $$e=\sqrt{1-\frac{b^2}{a^2}}$$ PS = e.PM Foci: $$(\pm ae, 0)$$ Directrix: $$x=\pm\frac{a}{e}$$ $$(a\cos\theta,b\sin\theta)$$ Tangent to Ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ [...]

7 02, 2017

Harder 3 Unit Topics

By |2017-03-11T11:41:34+10:00February 7th, 2017|Tags: , |0 Comments

Harder 3 Unit Topics Go to the Harder 3 Unit Topics forum to see more questions or ask your own Topics Covered in Harder 3 Unit Topics Inequalities Induction Geometry of Circle   Formulas Used in Harder 3 [...]

7 02, 2017

Volumes

By |2017-02-07T18:36:19+10:00February 7th, 2017|Tags: , |0 Comments

Volumes Go to the Volumes forum to see more questions or ask your own   Topics Covered in Volumes Motion in One Dimension Formulas Used in Volumes       Volumes Forum [...]

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