Understanding ellipse equation

Let us learn about ellipse equation
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is perpendicular to the axis. An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant.

Formula for Ellipse Equation

An ellipse with center at the origin (0, 0), is the graph of
With a > b > 0


An equation of an ellipse is generally defined as a conic obtained on slicing across obliquely one nappe of a cone. It is having two focus but parabols is having only one focus.It has the eccentricity less than one. If the eccentricity(e not <>the conic section like parabola or hyperbola. The ellipse's equation is represented by = 1 where b² = a² (1 − e²)

Also get help with How to solve Linear equation in one variable

What is the derivative of ln

Introduction

Derivative of In in mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula Where f ′ is the derivative of f. .


When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln (f); or, the derivative of the natural logarithm off. This follows directly from the chain rule.

Ratios

Definition of Ratios
Ratios means pairs of numbers and they are used to formulate comparisons of numbers. A ratio can be used to compares two numbers using a fraction. Ratios can be written three different ways:
If the ratio of one length to another is 1 : 2, this means that the second length is twice as large as the first. If a boy has 5 sweets and a girl has 3, the ratio of the boy's sweets to the girl's sweets is 5 : 3 . The boy has 5/3 times more sweets as the girl, and the girl has 3/5 as many sweets as the boy. Ratios behave like fractions and can be simplified.

Example

Simon made a scale model of a car on a scale of 1 to 12.5 . The height of the model car is 10cm.
(a) Work out the height of the real car.
The ratio of the lengths is 1 : 12.5 . So for every 1 unit of length the small car is, the real car is 12.5 units. So if the small car is 10 units long, the real car is 125 units long. If the small car is 10cm high, the real car is 125cm high.

(b) The length of the real car is 500cm. Work out the length of the model car.
We know that model : real = 1 : 12.5 . However, the real car is 500cm, so 1 : 12.5 = x : 500 (the ratios have to remain the same). x is the length of the model car. To work out the answer, we convert the ratios into fractions:

   

multiply both sides by 500:

500/12.5 = x

so x = 40cm
Example

Alix and Chloe divide £40 in the ratio 3 : 5. How much do they each get?

First, add up the two numbers in the ratio to get 8. Next divide the total amount by 8, i.e. divide £40 by 8 to get £5. £5 is the amount of each 'unit' in the ratio. To find out how much Alix gets, multiply £5 by 3 ('units') = £15. To find out how much Chloe gets, multiply £5 by 5 = £25.
Map Scales

If a map has a scale of 1 : 50 000, this means that 1 unit on the map is actually 50 000 units across the land. So 1cm on the map is 50 000cm along the ground (= 0.5km). So 1cm on the map is equivalent to half a kilometre in real life.
For 1 : 25 000, 1 unit on the map is the same as 25 000 units on the land. So 1 inch on the map is 25 000 inches across the land, or 1cm on the map is 25 000 cm in real life. You can manipulate these ratios if necessary.