Definitions [10]
A straight line drawn perpendicular to the axis and terminating at both ends of the curve is a double ordinate of the conic section.
The points of intersection of the conic section and the axis are called the vertices of the conic section.
The chord passing through the focus and perpendicular to the axis is called the latus rectum of the conic section.
A chord of a conic passing through the focus is called a focal chord.
A conic section is the locus of a point such that the ratio of its distance from a fixed point (focus) to a fixed line (directrix) is constant.
The point which bisects every chord of the conic passing through it is called the centre of the conic section.
The straight line passing through the focus and perpendicular to the directrix is called the axis of the conic section.
A parabola is the locus of a point which moves in a plane such that its distance from a fixed point (i.e. focus) is always equal to its distance from a fixed straight line (i.e. directrix).
An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is constant.
A hyperbola is the locus of a point in a plane which moves in such a way that the ratio of its distance from a fixed point (i.e. focus) to its distance from a fixed line (i.e. directrix) is always constant and greater than unity.
Formulae [1]
$$e = \frac{\text{distance from focus}}{\text{distance from directrix}}$$
Key Points
| Property | y² = 4ax | y² = −4ax | x² = 4ay | x² = −4ay |
|---|---|---|---|---|
| Vertex | (0, 0) | (0, 0) | (0, 0) | (0, 0) |
| Focus | (a, 0) | (−a, 0) | (0, a) | (0, −a) |
| Directrix | x + a = 0 | x − a = 0 | y + a = 0 | y − a = 0 |
| Axis | y = 0 | y = 0 | x = 0 | x = 0 |
| Axis of Symmetry | X-axis | X-axis | Y-axis | Y-axis |
| Eccentricity | 1 | 1 | 1 | 1 |
| Latus Rectum Length | 4a | 4a | 4a | 4a |
| Endpoints of Latus Rectum | (a, ±2a) | (−a, ±2a) | (±2a, a) | (±2a, −a) |
| Equation of Latus Rectum | x = a | x = −a | y = a | y = −a |
| Tangent at Vertex | x = 0 | x = 0 | y = 0 | y = 0 |
| Parametric Equations | x = at², y = 2at | x = −at², y = 2at | x = 2at, y = at² | x = 2at, y = −at² |
| Parametric Point | (at², 2at) | (−at², 2at) | (2at, at²) | (2at, −at²) |
| Focal Distance of P(x₁,y₁) | x₁ + a | a − x₁ | y₁ + a | a − y₁ |
| Fundamental Terms | Horizontal Ellipse (a>b) | Vertical Ellipse (a<b) |
|---|---|---|
| Equation | \[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\] | \[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\] |
| Centre | (0,0) | (0,0) |
| Vertices | (±a,0) | (0,±b) |
| Length of major axis | 2a | 2b |
| Length of minor axis | 2b | 2a |
| Foci | (±ae,0) | (0, ±be) |
| Relation between (a,b,e) | \[\mathrm{b}^{2}=\mathrm{a}^{2}(1-\mathrm{e}^{2})\] | \[\mathbf{a}^{2}=\mathbf{b}^{2}(1-\mathbf{e}^{2})\] |
| Eccentricity | \[\mathrm{e}=\frac{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}}{\mathrm{a}}\] | \[\mathrm{e}=\frac{\sqrt{\mathrm{b}^{2}-\mathrm{a}^{2}}}{\mathrm{b}}\] |
| Equation of directrices | \[x=\pm\frac{\mathrm{a}}{\mathrm{e}}\] | \[y=\pm\frac{b}{e}\] |
| Distance between foci | 2ae | 2be |
| Distance between directrices | \[\frac{2a}{e}\] | \[\frac{2b}{e}\] |
| Length of latus rectum | \[\frac{2\mathrm{b}^2}{a}\] | \[\frac{2\mathrm{a}^2}{b}\] |
| Endpoints of the latus rectum | \[\left(\pm ae,\pm\frac{b^{2}}{a}\right)\] | \[\left(\pm\frac{a^{2}}{b},\pm be\right)\] |
| Equation of axes | Major: (y = 0), Minor: (x = 0) | Major: (x = 0), Minor: (y = 0) |
| Parametric equations | \[\begin{cases} x=a\cos\alpha \\ y=b\sin\alpha & \end{cases}\] | \[\begin{cases} x=a\cos\alpha \\ y=b\sin\alpha & \end{cases}\] |
| Focal distances | \[\mid SP\mid=\left(a-ex_{1}\right)\mathrm{and}\mid S^{\prime}P\mid=\left(a+ex_{1}\right)\] | \[\mid SP\mid=(b-ey_{1})\mathrm{~and}\mid S^{\prime}P\mid=(b+ey_{1})\] |
| Sum of focal radii | 2a | 2b |
| Equation of the tangent at the vertex | (x = ± a) | (y = ± b) |
| Property |
Standard Hyperbola \[\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\] |
Conjugate Hyperbola \[\frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}=1\] |
|---|---|---|
| Centre | (0, 0) | (0, 0) |
| Vertices | (±a, 0) | (0, ±b) |
| Transverse Axis Length | 2a | 2b |
| Conjugate Axis Length | 2b | 2a |
| Foci | (±ae, 0) | (0, ±be) |
| Eccentricity | \[\mathrm{e}=\frac{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}{\mathrm{a}}\] | \[\mathrm{e}=\frac{\sqrt{\mathrm{b}^2+\mathrm{a}^2}}{\mathrm{b}}\] |
| Relation | \[\mathbf{b}^2=\mathbf{a}^2(\mathbf{e}^2-1)\] | \[\mathbf{a}^2=\mathbf{b}^2(\mathbf{e}^2-1)\] |
| Directrices | \[x=\pm\frac{\mathrm{a}}{\mathrm{e}}\] | \[y=\pm\frac{\mathrm{b}}{\mathrm{e}}\] |
| Length of Latus Rectum | \[\frac{2\mathrm{b}^2}{\mathrm{a}}\] | \[\frac{2\mathrm{a}^2}{\mathrm{b}}\] |
| Ends of Latus Rectum | \[\left(\pm ae,\pm\frac{b^{2}}{a}\right)\] | \[\left(\pm\frac{a^{2}}{b},\pm e\right)\] |
| Distance between Foci | 2ae | 2be |
| Difference of Focal Radii | 2a | 2b |
| Axis Equations | Transverse: y = 0, Conjugate: x = 0 | Transverse: x = 0, Conjugate: y = 0 |
| Parametric Equations | x = a secθ, y = b tanθ | x = a tanθ, y = b secθ |
| Parametric Point | (a secθ, b tanθ) | (a tanθ, b secθ) |
| Tangent at Vertex | x = ±a | y = ±b |
| Conic sections | Eccentricity | Equation of the curve | Equation of the tangent at a point (x₁, y₁) on it | Equation of tangents with slope m | Point of contact | Condition of tangency |
|---|---|---|---|---|---|---|
| Circle | – | x² + y² = a² | xx₁ + yy₁ = a² | \[y=\mathrm{mx}\pm\sqrt{\mathrm{a}^{2}\mathrm{m}^{2}+\mathrm{a}^{2}}\] | \[\left(\frac{-\mathbf{a}^2\mathbf{m}}{\mathbf{c}},\frac{\mathbf{a}^2}{\mathbf{c}}\right)\] | c² = a²m² + a² |
| Parabola | e = 1 | y² = 4ax | yy₁ = 2a(x + x₁) | \[y=\mathrm{m}x+\frac{\mathrm{a}}{\mathrm{m}}\] | \[\left(\frac{\mathrm{a}}{\mathrm{m}^{2}},\frac{2\mathrm{a}}{\mathrm{m}}\right)\] | \[c=\frac{a}{m}\] |
| Ellipse | 0 < e < 1 | \[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\] (a > b) | \[\frac{xx_{1}}{a^{2}}+\frac{yy_{1}}{b^{2}}=1\] | \[y=\mathrm{mx}\pm\sqrt{\mathrm{a}^{2}\mathrm{m}^{2}+\mathrm{b}^{2}}\] | \[\left(\frac{-a^{2}m}{c},\frac{b^{2}}{c}\right)\] | c² = a²m² + b² |
| Hyperbola | e > 1 | \[\frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{b}^2}=1\] | \[\frac{xx_1}{a^2}-\frac{yy_1}{b^2}=1\] | \[y=\mathrm{mx}\pm\sqrt{\mathrm{a}^{2}\mathrm{m}^{2}-\mathrm{b}^{2}}\] | \[\left(\frac{-\mathbf{a}^{2}\mathbf{m}}{\mathbf{c}},\frac{-\mathbf{b}^{2}}{\mathbf{c}}\right)\] | c² = a²m² − b² |
| Curve | Equation of auxiliary circle | Equation of the director circle |
|---|---|---|
| x² + y² = a² (circle) | - | x² + y² = 2a² |
| \[\frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{b}^2}=1\] (a > b) |
x² + y² = a² | x² + y² = a² + b² |
| \[\frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{b}^2}=1\] (a > b) |
x² + y² = a² | x² + y² = a² − b² |
