Conic section formulas: Circle, Ellipse, Parabola, Hyperbola Maths Class 11

Question

Study here Conic section formulas: Circle, Ellipse, Parabola, Hyperbola Maths Class 11 to a better understanding of this chapter.

Answer 1

Conic Section Formulas:

Since we have read simple geometrical figures in earlier classes. We already know about the importance of geometry in mathematics. Here we will learn conic section formulas. Circles, ellipses, parabolas and hyperbolas are in fact, known as conic sections or more commonly conics. As they can be obtained as intersections of any plane with a double-napped right circular cone. In fields such as planetary motion, design of telescopes and antennas, reflectors in flashlights and automobile headlights, etc. these curves have a very wide range of applications.

Circle:

To know more about circle visit Circle Formula.

Standard Equation of Circle:
x² + y² = r²
Where r is radius of circle.

Parabola:

It’s interesting to know that (‘Para’ means ‘for’ and ‘bola’ means ‘throwing’, i.e., the shape described when you throw a ball in the air). Conic section formulas for the parabola is listed below.

Equation of Parabola:

S. No.	Standard equation of Parabola	Focus	Directrix	Parametric equation of parabola
1	y2 = 4ax	(a, 0)	x = -a	x =at2 y =2at
2	y2 = - 4ax	(-a, 0)	x = a
3	x2 = 4ay	(0, a)	y = a
4	x2 = - 4ay	(0, -a)	y = - a

• Latus rectum of parabola: 4a

Ellipse:

The set of every point in a plane, the sum of whose distances from two fixed points in the plane is a constant. Conic section formulas for Ellipse is listed below.

Equation of Ellipse:

S. No.	Standard Equation of Ellipse	Parametric Equation of ellipse	F1 & F2
1	\(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1	x = a cos t y= b sin t	if a ≥ b ⇒ \(F_1\:(-\sqrt{a^2+b^2},0)\) \(F_2\:(-\sqrt{a^2+b^2},0)\)  if a < b \(F_1\:(0, -\sqrt{a^2+b^2})\) \(F_2\:(0, -\sqrt{a^2+b^2})\)
2	\(\frac{x^{2}}{b^{2}}\) + \(\frac{y^{2}}{a^{2}}\) = 1

• Hyperbola:Eccentricity of ellipse (e) = \(\frac{c}{a}\) = \(\frac{\sqrt{a^2-b^2}}{a}\)
• Latus rectum of ellipse (l) = \(\frac{b^{2}}{a}\)
• Area of Ellipse = π⋅a⋅b

The full set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant is Hyperbola. Conic section formulas for hyperbola is listed below.

Equation of Hyperbola:

S. No.	Standard equation of Hyperbola	Parametric equations of the Hyperbola:	F1 & F2
1	\(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1	x = \(\frac{a}{sin\:t}\) y = \(\frac{b\:sin\:t}{cos\:t}\)	if a ≥ b ⇒ \(F_1\:(-\sqrt{a^2+b^2},0)\) \(F_2\:(-\sqrt{a^2+b^2},0)\)  if a < b \(F_1\:(0, -\sqrt{a^2+b^2})\) \(F_2\:(0, -\sqrt{a^2+b^2})\)
2	\(\frac{y^{2}}{a^{2}}\) - \(\frac{x^{2}}{b^{2}}\) = 1

• Eccentricity of Hyperbola ( e ) = \(\frac{c}{a}\)
  Also, c ≥ a, the eccentricity is never less than one.
• Distance of focus from centre: ae
• Equilateral hyperbola: Hyperbola in which a = b
• Conic section formulas for latus rectum in hyperbola: \(\frac{2b^{2}}{a}\)

Conic section formulas examples:

1. Find an equation of the circle with centre at (0,0) and radius r.
   Solution:
   Here h = k = 0. Therefore, the equation of the circle is
   x² + y²= r²
2. Find the coordinates of the focus, axis, the equation of the directrix and latus rectum of the parabola y² = 16x.
   Solution:
   In this equation, y² is there, so the coefficient of x is positive so the parabola opens to the right. Comparing with the given equation y² = 4ax, we find that a = 4.
   Thus, the focus of the parabola is (4, 0) and the equation of the directrix of the parabola is x = – 4
   Length of the latus rectum is 4a = 4 × 4 = 16.