A LITTLE OF NUMBERS

Pending of a line

The slope of a line is the tangent of the angle that forms the line with the positive direction of the abscissa axis. Let P1 (x1; y1) and (x2; y2), P2 be two points on a line, not parallel to the Y axis; slope

If the slope (m) is greater than 0 it is said that the slope is positive, if the slope is less than 0 it is said that the slope is negative, if the slope is equal to 0 the line is parallel to the axis (x) of the Cartesian plane, and if the slope is undefined the line is parallel to the axis (y) of the Cartesian plane

EXAMPLE:

The high school stairs are 22 cm long and 15 cm high. Calculate the slope of the stairs and their equation.

(-22,0) and  (0,15)

P(0,15)

The slope of the stairs is  and the equations are 22x-13 + 225 = 0 and y = 22/15 x + 15

Circumference

The circumference is the geometric place of all points in the plane that are equidistant from a fixed point.

The fixed point is called CIRCUMFERENCE CENTER; Any line segment whose extreme points are the center of the circle and any point on it is called RADIO.

Equations of the circumference

Ordinary equation

At the origin: x 2 + y 2 = r 2

Outside the origin: (x ─ h) 2 + (y ─ k) 2 = r 2

General equation: x 2 + y 2 + Dx + Ey + F = 0

EXAMPLE:

In a source there are some discs that are above the source. One of them measures 52cm in diameter and its center is in the center and passes through the point (26.0)

Find the equation of the circle

 C(0,0)              r=26

The source has a radius of 26cm and its equation is x2+y2=676

Parable

It is defined as the geometric place of all the points in the plane that are equidistant from a fixed line called a guideline and a fixed point called a focus.

The focal axis is the axis perpendicular to the guideline that passes through the focus. It is the axis of symmetry of the parabola.

The point of the parabola that belongs to the focal axis is called the vertex.

Equations of the parable

With vertex at the origin and focus on the point (a, 0)

Horizontal y2 = 4ax

Vertical x2 = 4ay

Displaced

Horizontal (y – k) 2 = 4a (x – h)

Vertical (x – h) 2 = 4a (y – k)

General: Ax2 + By2 + Cx + Dy + E = 0

EXAMPLE:

The arc of the parabola measures 32m high at the vertex, and this parabola passes through the points (-4.27) and (4.27). Calculate the opening of the parabola.

(x-h)² =4p (y-k)           v(0,32)

x² = 4p(y-32)              P(4,27)    

(4)² = 4p(27-32)

16=4p(-5)

The opening of the parabola is equal to      

Ellipse

It is the geometric place of all points in the Cartesian plane, such that the sum of its distance to two fixed points is constant. Which is always greater than the distance between said fixed points.

Equations of the ellipse

Ordinary

Vertical x2 / a2 + y2 / b2 = 1

Horizontal x2 / b2 + y2 / a2 = 1

With center outside the origin

Vertical (x-h) 2 / a2 + (y-k) 2 / b2 = 1

Horizontal (x-h) 2 / b2 + (y-k) 2 / a2 = 1

General: Ax2 + Cy2 + Dx + Ey + F = 0

EXAMPLE:

Obtain the value of the axes, vertices and graph of the window in the form of an ellipse

16x2+25y2=400m-----16x2400+25y2400=400400

Canonical form: :      x225+y216=1

a2=25           b2=16            c2=a2-b2          c=9 

a=25          b=16           c2=(5)2-42           c=3  

a=5m            b=4m               c2=25-16

1. V±a,0---V(±5,0)
2. F±c,0---F(±3,0

3. Major axis length = 2nd ----- 10m

4. Minor axis length = 2b ----- 8m