Conic Sections: Definition, Formula, Equations & Solved Examples

Geometric figures like Circle, Hyperbola, Parabola, and Ellipse are referred to as conic sections because they are formed due to the intersection of a plane and a cone.

What is a Circle?

A circle is a figure in which every point is equidistant from the centre. The distance from the centre to any of its points is called a radius.

In general, a circle can be expressed by the following equation:

x² + y² + 2gx + 2fy + c = 0

Here, g, c, and f are constants, and the centre of the circle is (-g, -f). The radius of the circle here is r = square root of ( g² + f² - c).

If a circle passes through the origin, then the equation becomes x² + y² + 2gx + 2fy.

Also Check:  NCERT Solutions | NCERT Class 11 Maths Solutions

What is a Parabola?

A parabola is a curve-like figure on which any point is equidistant from a fixed point called the focus and a straight line that is also fixed, called the directrix.

A parabola has 4 forms viz. y² = 4ax, y² = -4ax, x² = 4ay, and x² = -4ay.

What is an Ellipse?

An ellipse looks like a skewed circle and is referred to as a point set wherein the sum of all the points remains constant from 2 fixed points. The standard forms of an ellipse are given below:

(x²/a²) + (y²/b²) = 1 and (x²/b²) + (y²/a²) = 1.

In both these forms, a > b and b² = a²(1 - e²) where e > 1.

What is a Hyperbola?

A hyperbola is an open curve obtained from the intersection of a circular conic section with a plane. Here, the ratio of the distance of the points remains constant from a point called the focus and a line called the directrix. The standard forms of a Hyperbola are given below:

(x²/a²) - (y²/b²) = 1 and (y²/a²) - (x²/b²) = 1.

Check here the general equations for different types of conic sections.

Important Topics:

All the conic sections' topics are extensively covered in Class XI and carry a weightage of 4 to 7 marks. It includes MCQ (Multiple Choice Questions), fill in the blanks, short and long answer questions.

Various applications of conic sections are as follows:

• The planet orbits in an elliptical shape.
• Telescopes, satellites, and car headlights are in parabolic
• A hyperbola is used in navigation systems, optics, and acoustics

Related Topics:

The conic sections miscellaneous exercise consists of 10 questions. Students can check the NCERT Class 11 Maths textbook for miscellaneous exercise questions. Solving the Conic Sections class 11 miscellaneous exercise will help to know how well you have understood the topics. Seek the teacher's help if unable to solve the question, and regularly practice the exercise questions. Also, students who are looking for solutions to miscellaneous exercises can check the NCERT conic section class 11 solutions. 

1. Calculate the equation of the circle with the centre at (0, 3) and radius 2.

Solution. The equation of the circle will be (x - 0)² + (y - 3)² = (2)²

 x² + y² - 4y + 4 = 4

 x² + y²- 4y = 0.

2. The equation of the parabola is y² = 20x. Find its focus, latus rectum's length, its axis and equation of the directrix.

Solution. From the general equation y² = 4ax we get a = 5. Therefore, the length of the latus rectum will be 4a = 4 x 5 = 20.

The coordinate of the focus will be (5, 0), and the equation of the directrix will be x = -5.

The parabola's axis will be y = 0.

3. Find the equation of the ellipse with the centre at the origin and major axis falling on the y-axis going through points (3, 2) and (1, 6).

Solution. The centre is at (0, 0) and the equation of this ellipse is of the below form:

(x²/b²) + (y²/a²) = 1

As it passes through the points (3, 2) and (1, 6) this equation can be written as:

(9/b²) + (4/a²) = 1

Therefore, (1/b²) + (36/a²) = 1

From the above equations we get the values a² = 40b² = 10

Therefore, finally the equation becomes,

(x²/10) + (y²/40) = 1.

Q: Is every circle an ellipse?

A: Yes, every circle is an ellipse because a circle is a special case of an ellipse with the same distance from the centre for all the points.
Q: Give a real-life example of an ellipse.

A: Some of the real-life examples of ellipses are:

• Earth revolves around the sun in an elliptical orbit.
• Car headlights are designed in a parabolic shape.
• The path followed by a ball when thrown is a parabolic trajectory.

Q: Give a real-life example of a hyperbola.

A: The example of a hyperbola is:

• Cooling towers in power plants are designed in a hyperbolic shape for strength and airflow.
• It is used in navigation systems to find locations.

Q: Give a real-life example of a parabola.

A:

• The dish of a satellite or TV, and car headlights are designed in a parabolic shape to focus signals or light.
• The trajectory of the ball is in a parabolic shape.