# Mathematical Analytic Geometry

Mathematical Analytic Geometry is a coordinate system or branch of algebra that is used to model geometric objects like points, straight lines and circles angles in the Cartesian plane, the geometry shapes are defined as using coordinate system or algebric principles.

In this plane analytic geometry where points are represent as pairs of numbers like (x,y) and straight lines are defined as sets of points.

- A and B is two points and the distance between two points
- P = √( (x
_{2}– x_{1}) + ( y_{2}– y_{1}))^{2}

- P = √(x
^{2}+ y^{2})

### Triangle Analytic Geometry

Area of Triangle with vertices at A, B, C

- A = ± 1 / 2 [(x
_{2}– x_{1})(y_{3}– y_{1})(x_{3}– x_{1})(y_{2}– y_{1})] - = ± 1 / 2 [x
_{1}(y_{2}– y_{3}) + x_{2}(y_{3}– y_{1}) + x_{3}(y_{1}– y_{2})]

### Line Of Equation – Mathematical Analytic Geometry

- A, B two points joining
- y – y
_{1}/ y_{2}– y_{1}= x – x_{1}/ x_{2}– x_{1}

**Passing point A and parallel with line Y = ax + b**

**Passing A point and perpendicular with line** Y = ax + b

- y – y
_{0}= -1 / a (x – x_{0})

### Equation Of Circle – Mathematical Analytic Geometry

Radius of center is the form of the circle equation where circle with radius r and the center at (a,b)

- (x – a)
^{2}+ (y – b)^{2}= r^{2}

The circle with center at origin

- x
^{2}+ y^{2}= r^{2}

### Ellipse – Mathematical Analytic Geometry

**Equation Of Ellipse :**

**Horizontal Major Axis :**

x^{2}/ a^{2}+ y^{2}/ b^{2}= 1

**Vertical Major Axis :**

x^{2}/ b^{2}+ y^{2}/ a^{2}= 1

**Eccentricity Ellipse :**

ε = c / a = √(a^{2} – b^{2})/ a < 1

**Area of Ellipse :**A = πab

### Hyperbola

**Equation of Hyperbola :**x^{2}/ a^{2}– y^{2}/ b^{2}= 1**Eccentricity Of Hyperbola :**ε = c / a = √( a^{2}+ b^{2})/ a > 1

### Parabola

**Area of Parabola :**A = 2 / 3 lc**Equation of Parabola :**y^{2}= 2px**Eccentricity of Parabola :**ε = FM / MK = 1

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