Conic Sections: Overview, Questions, Preparation

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 an origin then the equation becomes x² + y² + 2gx + 2fy.

What is a Parabola?

A parabola is a curve-like figure on which any point is equidistant from a fixed point called 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 the circular conic section with a plane. Here, the ratio of the distance of the points remains constant from a point called focus and a line called the directrix. The standard forms of Hyperbola are given below:

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

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.

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: What are the applications of conic sections?

Q: Is every circle an ellipse?

Q: Give a real-life example of an ellipse.

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

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