Quick Revision Notes: Class 11 Mathematics

Chapter: Conic Sections (शंकु परिच्छेद)

---

Introduction to Conic Sections

A conic section is the locus of a point moving in a plane such that the ratio of its distance from a fixed point (Focus) to its fixed line (Directrix) is always a constant, known as the Eccentricity ($e$).

• If $e = 1$: Parabola (परवलय)
• If $e < 1$: Ellipse (दीर्घवृत्त)
• If $e > 1$: Hyperbola (अतिपरवलय)
• If $e = 0$: Circle (वृत्त)

---

1. Circle (वृत्त)

A circle is the set of all points in a plane that are equidistant from a fixed point (center) in the plane.

• Standard Equation: (x - h)^2 + (y - k)^2 = r^2 (where $(h, k)$ is the center and $r$ is the radius)
• Equation with Center at Origin $(0, 0)$: x^2 + y^2 = r^2
• General Equation: x^2 + y^2 + 2gx + 2fy + c = 0
  • Center: (-g, -f)
  • Radius: r = sqrt(g^2 + f^2 - c)

---

2. Parabola (परवलय)

A parabola is the set of all points in a plane that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line.

Standard Forms of Parabola:

Property	Type I: $y^2 = 4ax$	Type II: $y^2 = -4ax$	Type III: $x^2 = 4ay$	Type IV: $x^2 = -4ay$
Axis of Symmetry	X-axis	X-axis	Y-axis	Y-axis
Coordinates of Focus	$(a, 0)$	$(-a, 0)$	$(0, a)$	$(0, -a)$
Equation of Directrix	$x = -a$	$x = a$	$y = -a$	$y = a$
Length of Latus Rectum	$4a$	$4a$	$4a$	$4a$
Equation of Latus Rectum	$x = a$	$x = -a$	$y = a$	$y = -a$

• Eccentricity ($e$): For all parabolas, $e = 1$.

---

3. Ellipse (दीर्घवृत्त)

An ellipse is the set of all points in a plane whose distances from two fixed points (foci) in the plane have a constant sum.

• Standard Equations:
  1. x^2/a^2 + y^2/b^2 = 1 (where $a > b$)
  2. x^2/b^2 + y^2/a^2 = 1 (where $a > b$)

Key Terms & Formulas for x^2/a^2 + y^2/b^2 = 1 (when $a > b$):

• Eccentricity ($e$): e = sqrt(1 - b^2/a^2) (where $b^2 = a^2(1 - e^2)$)
• Foci: $(\pm ae, 0)$
• Vertices: $(\pm a, 0)$
• Length of Major Axis: $2a$
• Length of Minor Axis: $2b$
• Equation of Directrices: x = \pm a/e
• Length of Latus Rectum: 2b^2 / a

---

4. Hyperbola (अतिपरवलय)

A hyperbola is the set of all points in a plane whose distances from two fixed points in the plane have a constant difference.

• Standard Equations:
  1. x^2/a^2 - y^2/b^2 = 1 (Transverse axis along X-axis)
  2. y^2/a^2 - x^2/b^2 = 1 (Transverse axis along Y-axis)

Key Terms & Formulas for x^2/a^2 - y^2/b^2 = 1:

• Eccentricity ($e$): e = sqrt(1 + b^2/a^2) (Note: $e > 1$ for hyperbola)

• Relation between $a, b, c$: c^2 = a^2 + b^2 (where $c = ae$ is the distance of focus from center)

• Foci: $(\pm c, 0)$ or $(\pm ae, 0)$

• Vertices: $(\pm a, 0)$

• Length of Transverse Axis: $2a$

• Length of Conjugate Axis: $2b$

• Equation of Directrices: x = \pm a/e

• Length of Latus Rectum: 2b^2 / a

• Special Case - Rectangular Hyperbola: If $a = b$, the hyperbola is called a rectangular hyperbola, and its eccentricity is always sqrt(2).

---

Quick Tip for MP Board Exams:

• Always draw a rough sketch of the conic section before solving problems based on foci, vertices, and latus rectum.
• Memorize the relationship between $a$, $b$, and $e$ for Ellipse (b^2 = a^2(1 - e^2)) and Hyperbola (b^2 = a^2(e^2 - 1)).