JEE Main - Conic sections: parabola ellipse and hyperbola in standard forms E-Book Study Material

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Conic Sections: Parabola, Ellipse, and Hyperbola in Standard Forms

Introduction to Conic Sections

Conic sections are curves obtained by the intersection of a plane with a double cone. Depending on the angle of the plane relative to the cone, we get different types of curves: a circle, an ellipse, a parabola, or a hyperbola. Special cases can also result in a point, a line, or a pair of intersecting lines.

The standard forms of these curves are fundamental to understanding their properties, such as focus, directrix, center, vertices, and asymptotes. These properties are crucial for solving various problems in geometry, physics (like planetary orbits), and engineering.

Parabola

A parabola is defined as the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Standard Forms of a Parabola

The standard forms depend on the position of the focus and the directrix relative to the coordinate axes. The vertex is usually considered the origin (0,0) in these standard forms.

1. Parabola with Vertex at Origin and Axis along the x-axis

This is the most common form. The axis of symmetry is the x-axis.

Equation: y² = 4ax
Focus: (a, 0)
Directrix: x = -a
Vertex: (0, 0)
Axis of Symmetry: x-axis (y=0)
Eccentricity (e): 1
Latus Rectum: The length of the chord through the focus perpendicular to the axis of symmetry. Its length is 4|a|.

If a > 0, the parabola opens to the right. If a < 0, the parabola opens to the left, and the equation is y² = -4|a|x. The focus is (-|a|, 0) and the directrix is x = |a|.

2. Parabola with Vertex at Origin and Axis along the y-axis

The axis of symmetry is the y-axis.

Equation: x² = 4ay
Focus: (0, a)
Directrix: y = -a
Vertex: (0, 0)
Axis of Symmetry: y-axis (x=0)
Eccentricity (e): 1
Latus Rectum: Length is 4|a|.

If a > 0, the parabola opens upwards. If a < 0, the parabola opens downwards, and the equation is x² = -4|a|y. The focus is (0, -|a|) and the directrix is y = |a|.

Parabola with Vertex at (h, k)

If the vertex is shifted to a point (h, k), the equations are modified by replacing x with (x-h) and y with (y-k).

Equation: (y-k)² = 4a(x-h) (Axis parallel to x-axis)
Focus: (h+a, k)
Directrix: x = h-a
Equation: (x-h)² = 4a(y-k) (Axis parallel to y-axis)
Focus: (h, k+a)
Directrix: y = k-a

Parabola Shortcut: For y² = 4ax, remember the focus is (a,0) and directrix is x=-a. For x² = 4ay, focus is (0,a) and directrix is y=-a. The '4a' term is key to the latus rectum length. If the squared term is 'y', the axis is the x-axis. If the squared term is 'x', the axis is the y-axis.

Ellipse

An ellipse is defined as the set of all points in a plane such that the sum of the distances from two fixed points (the foci) is constant.

Standard Forms of an Ellipse

The standard forms assume the center of the ellipse is at the origin (0,0).

1. Ellipse with Center at Origin and Major Axis along the x-axis

The major axis is longer than the minor axis.

Equation: x²/a² + y²/b² = 1, where a > b > 0.
Vertices: (±a, 0)
Co-vertices: (0, ±b)
Foci: (±c, 0), where c² = a² - b².
Center: (0, 0)
Major Axis Length: 2a
Minor Axis Length: 2b
Eccentricity (e): c/a (Always between 0 and 1 for an ellipse)
Latus Rectum: Length is 2b²/a.

2. Ellipse with Center at Origin and Major Axis along the y-axis

Here, the major axis is along the y-axis.

Equation: x²/b² + y²/a² = 1, where a > b > 0.
Vertices: (0, ±a)
Co-vertices: (±b, 0)
Foci: (0, ±c), where c² = a² - b².
Center: (0, 0)
Major Axis Length: 2a
Minor Axis Length: 2b
Eccentricity (e): c/a
Latus Rectum: Length is 2b²/a.

Note that in the standard form x²/p + y²/q = 1, the larger denominator indicates the direction of the major axis. If p > q, the major axis is along the x-axis. If q > p, the major axis is along the y-axis.

Ellipse with Center at (h, k)

Shifting the center to (h, k) modifies the equations:

Equation: (x-h)²/a² + (y-k)²/b² = 1 (Major axis parallel to x-axis)
Equation: (x-h)²/b² + (y-k)²/a² = 1 (Major axis parallel to y-axis)

For these shifted ellipses, the foci and vertices are determined by adding (h, k) to the coordinates of the foci and vertices of the ellipse centered at the origin.

Ellipse Shortcut: The relationship c² = a² - b² is crucial. Remember 'a' is always associated with the major axis, and 'b' with the minor axis. The foci lie on the major axis. For x²/a² + y²/b² = 1 with a > b, foci are (±c, 0). For x²/b² + y²/a² = 1 with a > b, foci are (0, ±c). Eccentricity e = c/a.

Hyperbola

A hyperbola is defined as the set of all points in a plane such that the absolute difference of the distances from two fixed points (the foci) is constant.

Standard Forms of a Hyperbola

The standard forms assume the center of the hyperbola is at the origin (0,0). A hyperbola has two branches.

1. Hyperbola with Center at Origin and Transverse Axis along the x-axis

The transverse axis is the line segment connecting the two vertices.

Equation: x²/a² - y²/b² = 1
Vertices: (±a, 0)
Foci: (±c, 0), where c² = a² + b².
Center: (0, 0)
Transverse Axis Length: 2a
Conjugate Axis Length: 2b
Eccentricity (e): c/a (Always greater than 1 for a hyperbola)
Asymptotes: The lines that the hyperbola approaches but never touches. For this form, the equations are y = ±(b/a)x.
Latus Rectum: Length is 2b²/a.

2. Hyperbola with Center at Origin and Transverse Axis along the y-axis

The transverse axis is along the y-axis.

Equation: y²/a² - x²/b² = 1
Vertices: (0, ±a)
Foci: (0, ±c), where c² = a² + b².
Center: (0, 0)
Transverse Axis Length: 2a
Conjugate Axis Length: 2b
Eccentricity (e): c/a
Asymptotes: y = ±(a/b)x.
Latus Rectum: Length is 2b²/a.

For hyperbolas, the equation x²/p - y²/q = 1 has its transverse axis along the x-axis (a²=p, b²=q). The equation y²/p - x²/q = 1 has its transverse axis along the y-axis (a²=p, b²=q). Note the difference in the relationship between a and b compared to ellipses: for hyperbolas, c² = a² + b².

Hyperbola with Center at (h, k)

Shifting the center to (h, k):

Equation: (x-h)²/a² - (y-k)²/b² = 1 (Transverse axis parallel to x-axis)
Equation: (y-k)²/a² - (x-h)²/b² = 1 (Transverse axis parallel to y-axis)

The foci, vertices, and asymptotes are found by shifting the corresponding elements of the hyperbola centered at the origin by the vector (h, k).

Hyperbola Shortcut: The key relationship is c² = a² + b². Remember 'a' is always associated with the positive term's denominator in the standard form. If x² is positive, the transverse axis is along the x-axis, and foci are (±c, 0). If y² is positive, the transverse axis is along the y-axis, and foci are (0, ±c). Asymptotes are lines the hyperbola approaches.

Summary Table of Standard Forms (Vertex/Center at Origin)

Conic Section	Equation	Foci	Vertices	Eccentricity (e)	Asymptotes (if applicable)
Parabola	y² = 4ax	(a, 0)	(0, 0)	1	N/A
Parabola	x² = 4ay	(0, a)	(0, 0)	1	N/A
Ellipse	x²/a² + y²/b² = 1 (a > b)	(±c, 0), c²=a²-b²	(±a, 0)	c/a (< 1)	N/A
Ellipse	x²/b² + y²/a² = 1 (a > b)	(0, ±c), c²=a²-b²	(0, ±a)	c/a (< 1)	N/A
Hyperbola	x²/a² - y²/b² = 1	(±c, 0), c²=a²+b²	(±a, 0)	c/a (> 1)	y = ±(b/a)x
Hyperbola	y²/a² - x²/b² = 1	(0, ±c), c²=a²+b²	(0, ±a)	c/a (> 1)	y = ±(a/b)x

Key Differences and Relationships

Understanding the differences in the equations and the relationships between a, b, and c is vital.

Parabola: Eccentricity is always 1. It has one focus and one directrix.
Ellipse: Eccentricity is between 0 and 1. It has two foci, and c² = a² - b², where a > b. The sum of distances from any point on the ellipse to the foci is constant (2a).
Hyperbola: Eccentricity is greater than 1. It has two foci, and c² = a² + b². The absolute difference of distances from any point on the hyperbola to the foci is constant (2a).

The general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 represents a conic section. The nature of the conic section can be determined by the discriminant B² - 4AC.

If B² - 4AC < 0, it's an ellipse (or circle if A=C and B=0).
If B² - 4AC = 0, it's a parabola.
If B² - 4AC > 0, it's a hyperbola.

This classification is useful when the conic section is not in a standard form, especially when there's an xy term indicating rotation. However, for JEE Main, focusing on the standard forms is usually sufficient for direct application problems.

Example Problems and Solutions

Example 1: Parabola

Find the focus and directrix of the parabola y² = 12x.

Solution: This equation is in the standard form y² = 4ax. Comparing, we get 4a = 12, so a = 3. The focus is at (a, 0), which is (3, 0). The directrix is x = -a, which is x = -3.

Example 2: Ellipse

Find the length of the major axis, foci, and eccentricity of the ellipse 4x² + 9y² = 36.

Solution: First, convert the equation to standard form by dividing by 36: x²/9 + y²/4 = 1. Here, a² = 9 and b² = 4. Since a² > b², the major axis is along the x-axis. a = 3 and b = 2. Length of the major axis = 2a = 2(3) = 6. Now, find c using c² = a² - b² = 9 - 4 = 5. So, c = √5. The foci are at (±c, 0), which are (±√5, 0). Eccentricity e = c/a = √5 / 3.

Example 3: Hyperbola

Find the vertices, foci, and equations of the asymptotes for the hyperbola x²/16 - y²/9 = 1.

Solution: This is in the standard form x²/a² - y²/b² = 1. Here, a² = 16 and b² = 9. a = 4 and b = 3. The transverse axis is along the x-axis. Vertices are at (±a, 0), which are (±4, 0). Find c using c² = a² + b² = 16 + 9 = 25. So, c = 5. Foci are at (±c, 0), which are (±5, 0). The equations of the asymptotes are y = ±(b/a)x, which are y = ±(3/4)x.

Exam Tip: Always check which variable (x or y) is associated with the positive term in the standard equation of an ellipse or hyperbola. This determines the orientation of the major/transverse axis. For ellipses, a > b and c² = a² - b². For hyperbolas, c² = a² + b² and 'a' is linked to the positive term.

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