Class 11 Maths • Chapter 10 • Comprehensive Interactive Notes
Conic sections are curves obtained by intersecting a right circular cone with a plane.
Definition by Eccentricity (\(e\)): The locus of a point which moves such that the ratio of its distance from a fixed point (Focus) to its perpendicular distance from a fixed line (Directrix) is constant.
Drag the slider to change 'e' and see the shape.
A circle is the set of all points in a plane equidistant from a fixed point (center).
Equation: \( (x-h)^2 + (y-k)^2 = r^2 \)
Locus of a point equidistant from focus and directrix.
| Equation | Axis | Focus | Directrix |
|---|---|---|---|
| \( y^2 = 4ax \) | X-axis (Right) | \( (a, 0) \) | \( x = -a \) |
| \( y^2 = -4ax \) | X-axis (Left) | \( (-a, 0) \) | \( x = a \) |
| \( x^2 = 4ay \) | Y-axis (Up) | \( (0, a) \) | \( y = -a \) |
| \( x^2 = -4ay \) | Y-axis (Down) | \( (0, -a) \) | \( y = a \) |
For \( y^2 = 4ax \) or similar.
| Conic | Parametric Point |
|---|---|
| Parabola | \( (at^2, 2at) \) |
| Ellipse | \( (a\cos\theta, b\sin\theta) \) |
| Hyperbola | \( (a\sec\theta, b\tan\theta) \) |
Why CBSE asks this:
Helps simplify tangent, normal & distance problems.
Tangent to Parabola \( y^2 = 4ax \):
\[ ty = x + at^2 \]
Normal to Parabola:
\[ y = -tx + 2at + at^3 \]
Exam Note:
Mostly 3–4 mark questions.
e < 1
Standard Eq: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
*Assumes a > b > 0 for Horizontal Ellipse.
e > 1
Standard Eq: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
*Calculates for standard horizontal hyperbola.
Ellipse:
\[ \text{Director Circle: } x^2 + y^2 = a^2 + b^2 \]
Hyperbola:
\[ \text{Director Circle: } x^2 + y^2 = a^2 - b^2 \]
CBSE Tip:
These are asked directly or inside proofs.
Test your knowledge with these questions.
1. The eccentricity of a circle is:
2. For a parabola \( y^2 = -8x \), the focus is:
3. In an ellipse, the relationship between a, b, c is:
4. The length of Latus Rectum for \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is:
5. If \( e > 1 \), the conic is a:
Self-Test: