Conic Sections Class 11 – When a plane cuts the cone other than the vertex, we have the following situations:

(a) When \(\beta = 90^{0}\) , the section is a circle

(b) When \(\alpha < \beta < 90^{0}\), the section is an ellipse

(c) When \(\alpha = \beta\); the section is a parabola

(d) When \(0 \leq \beta < \alpha\); the section is a hyperbola

Where, β is the angle made by the plane with the vertical axis of the cone.

**Circle: **Set of points in a plane equidistant from a fixed point. A circle with radius r and centre (h, k) can be represented as \(\left ( x-h \right )^{2}+\left ( y-k \right )^{2}=r^{2}\)

**Parabola:** Set of points in a plane that are equidistant from a fixed line and point. A parabola with a > 0, focus at (a, 0), and directrix x = – a can be represented as \(y^{2}=4ax\)

In parabola \(y^{2}=4ax\), the length of latus rectum is given by 4a.

**Ellipse: **The sum of distances of set of points in a plane from two fixed points is constant. An ellipse with foci on the x-axis can be represented as: \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)

In ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), the length of the latus rectum is given by \(\frac{2b^{2}}{a}\).

**Hyperbola: **The difference of distances of set of points in a plane from two fixed points is constant. The hyperbola with foci on the x-axis can be represented as:

\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\)

In a Hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\), the length of latus rectum is given by \(\frac{2b^{2}}{a}\)

### Conic Sections Class 11 Examples