Solved Problems On Ellipse

The ellipse concept is a simple concept that can get learned with some simple ideas. Here, we will see some formulas related to the problems with the ellipse. And see how easily it can get solved. The ellipse word refers to the locus of the point in the present plane so that the distance sum will remain constant.

The fixed points are a singular focus in the plane, whereas the fixed line is the directrix. Mainly, the eccentricity of an ellipse is represented by ‘e.’ Go with more detail related to the problems of ellipses. The primary term is that the ellipse is as same as the conic section parts.

What do you mean by ellipse?

The meaning of an ellipse is in a plane’s locus points so that when we look at the sum of their distances, it is equal to its fixed points. Foci are considered as the setpoint, and it is mainly covered with curves. Here, the constant ratio is known as the ellipse’s eccentricity, whereas the fixed line is called the directrix. The ellipse’s present eccentricity is considered one factor of it, and it is represented by the symbol ‘e.’

• The ellipse is presented in a shape known as an oval shape.
• The Ellipse area is mainly considered and looks from its minor and major axis.
• The Ellipse area is equal to the πab.
• The formula πab, a & b are considered semi major’s & minor’s length.
• Ellipse is the same as hyperbola & parabola.

Parts of an ellipse

The ellipse is divided into various parts. Here, we will discuss the terms of the ellipse in detail. All the ellipse parts have their importance. Below is the list that contains the detailed items of the ellipse.

1. Centre:

• The centre is known as the midpoint of the line.
• It is the midpoint of the foci from which it is joining the line.

2. Focus:

• Mainly, there are two points of focus.
• One is f (c,o), and another one is f ( -c,0)
• And when we see the distance between the foci, it is always equal to 2a.

3. Minor axis:

• The minor axis of the ellipse is with the y-axis.
• The minor axis is considered the smallest term.
• The length is always equal to 2b.

4. Major axis:

• The term central axis of an ellipse is with the x-axis.
• The length of the central axis is always equal to 2a.

5. Transverse axis:

• When the line gets passed from the two foci & ellipse centres, it is known as the transverse axis.

6. Eccentricity:

• The constant ratio is known as the ellipse’s eccentricity, whereas the fixed line is called the directrix. 
• The ellipse’s present eccentricity is considered one factor of it, and it is represented by the symbol ‘e.’

7. Conjugate axis:

• When the line gets passed with the centre of the ellipse axis.
• And that is perpendicular to the transverse axis.

Properties of ellipse

Properties are determined by separating the different shapes of it. Here, we will see its various properties below:

• The ellipse consists of two main parts: the major axis and the minor axis.
• Mainly eccentricity is represented by the symbol ‘e.’

Solved problems

Question 1: What is the equation of an ellipse when its eccentricity is equals to 

Solution: The equation of ellipse is represented by the x,y variables. Here, the equation is x2/a2 with addition to y2/b2 that is equal to 1 i.e. x2/a2 + y2/b2 = 1.

Foci coordinates are (0,4)

be= 4

b = 5 

b2 = 25

 a2  = b2(1 – e2)

a2 = 25(1 – 16/25)

 a2 = 9

 Now, in the equation x2/9 + y2/25 = 1

Question 2: What is the area of the axis when the minor & major area of the axis is equal to the 14 & 8.

Solution: We have to find the area of an ellipse.

It is given that 2a = 14

Whereas a = 14/2 = 7

And 2b = 8

Whereas b = 8/2 = 4

Area of an ellipse = πab

Whereas π (7) (4) = 28π, 28 (22/7) = 88.

Conclusion

We can see that the main parts of the ellipse are similar to the parabola & hyperbola. The shape of the ellipse is present in the shape of an oval. When we talk about the formula of an ellipse, then it is equal to the πab. Whereas in πab, a & b are the length of the semi-major and semi-minor lengths. The parts of the ellipse get formed with the plane conic section. And when it cuts at the angle from its base. And when we talk about the circle, it gets formed with the intersection of the plane with the base.