Definitions [6]
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).
Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola in following fig.
To find the Length of the latus rectum of the parabola `y^2` = 4ax in following fig.
By the definition of the parabola, AF = AC.
But AC = FM = 2a
Hence AF = 2a.
And since the parabola is symmetric with respect to x-axis AF = FB and so
AB = Length of the latus rectum = 4a.
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.
Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse in following fig.
To find the length of the latus rectum of the ellipse `x^2/a^2 + y^2/b^2 = 1`
Let the length of `AF_2` be l.
Then the coordinates of A are (c, l ),i.e., (ae, l )
Since A lies on the ellipse `x^2/a^2 + y^2/b^2 = 1`,
`(ae)^2/a^2+l^2/b^2=1`
`=> l^2 = b^2(1-e^2)`
But `e^2 = c^2/a^2 = (a^2 - b^2)/a^2 = 1- b^2/a^2`
Therefore `l^2 = b^4/a^2, i.e., l = b^2/a`
Since the ellipse is symmetric with respect to y-axis ,`AF_2` = `F_2B` and so length of the latus rectum is `(2b)^2/a.`
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.
Equation of the form ax² + 2hxy + by² + 2gx + 2fy + c = 0 is called a general second-degree equation.
- The necessary conditions for a general second-degree equation to represent a pair of lines are: (i) abc + 2fgh − af² − bg² − ch² = 0, (ii) h² − ab ≥ 0
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 |
| Condition | Type of Lines |
|---|---|
| \[\Delta=0,h^2>ab\] | Intersecting lines |
| \[\Delta=0,h^2 = ab\] | Coincident lines |
| \[\Delta=0,h^2<ab\] | Imaginary lines |
| (\[\Delta=0,h^2=ab\] and \[bg^{2}=af^{2}\] | Parallel lines |
For Standard Circle: x² + y² = a²
| Sr. No. | Description | Formula |
|---|---|---|
| i. | Tangent at a point (x₁, y₁) | xx₁ + yy₁ = a² |
| ii. | Parametric form of tangent at P(θ) | x cosθ + y sinθ = a |
| iii. | Condition of tangency for the line y = mx + c | \[\mathrm{c=\pm a~\sqrt{1+m^{2}}}\] |
| Point of contact | \[\left(\frac{-\mathrm{a}^{2}\mathrm{m}}{\mathrm{c}},\frac{\mathrm{a}^{2}}{\mathrm{c}}\right)\] | |
| iv. | Equation of tangent in terms of its slope m | \[y=\mathrm{m}x\pm\mathrm{a}\sqrt{1+\mathrm{m}^{2}}\] |
| v. | Length of tangent from the point (x₁, y₁) | \[\sqrt{S_{1}}=\sqrt{x_{1}^{2}+y_{1}^{2}-a^{2}}\] |
| vi. | Equation of the Director circle | x² + y² = 2a² |
For General Circle: x² + y² + 2gx + 2fy + c = 0
| Sr. No. | Description | Formula |
|---|---|---|
| i. | Tangent at a point (x₁, y₁) | xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0 |
| ii. | Length of tangent from the point (x₁, y₁) | \[\sqrt{S_{1}}=\sqrt{x_{1}^{2}+y_{1}^{2}+2gx_{1}+2fy_{1}+c}\] |
Number of Common Tangents:
| Case | Diagram | No. of Tangents | Condition |
|---|---|---|---|
| Disjoint circles |  |
4 | d > r₁ + r₂ |
| Touch externally |  |
3 | d = r₁ + r₂ |
| Intersecting circles |  |
2 | d < r₁ + r₂ |
| Touch internally |  |
1 | d = \[\left|\mathbf{R}_{1}-\mathbf{R}_{2}\right|\] |
| Concentric circles |  |
0 | d = 0 |
Equation of a pair of tangents:
(x² + y² − a²)(x₁² + y₁² − a²) = (xx₁ + yy₁ − a²)²
Concepts [15]
- Sections of a Cone
- Conics as a Section of a Cone
- Definition of Foci, Directrix, Latus Rectum
- Parabola and its types
- Latus Rectum
- Ellipse and its Types
- Latus Rectum
- Hyperbola and its Types
- Transverse and Conjugate Axes
- Coordinates of Vertices
- Foci and Centre
- Equations of the Directrices and the Axes
- General Second Degree Equation
- Equation of Tangent and Condition of Tangency
- Point of Contact and Locus Problems
