Definitions [1]
Definition: Ellipse
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.
Key Points
Key Points: Ellipse and its Types
| 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) |
Key Points: Area Under Simple Curves
- \[\mathrm{A}=\int_{x=a}^{x=b}y\mathrm{d}x=\int_{a}^{b}\mathrm{f}(x)\mathrm{d}x\]
- If below x-axis → use |f(x)|
- Area w.r.t. Y-axis →\[\mathrm{A}=\int_{y=c}^{y=d}x\mathrm{d}y=\int_{c}^{d}\mathrm{f}(y)\mathrm{d}y\]
- If the curve crosses the axis → split the interval
Key Points: Area Under the Curve
| Case | Formula |
|---|---|
| Area under y = f(x) | \[\int_a^bf(x)dx\] |
| Area between curves | \[\int_a^b[f(x)-g(x)]dx\] |
| Area w.r.t. y-axis x = g(y) |
\[\int_c^dg(y)dy\] |
| Even function | \[2\int_0^af(x)dx\] |
| Odd function | 0 |
If the area A lies below the X-axis, then A is negative, and in this case, we take | A |.
Key Points: Symmetry of Curve
| Type of Symmetry | Condition | Replacement Rule | Result |
|---|---|---|---|
| About the X–axis | (x, y) ∈ C ⇔ (x, -y) ∈ C | Replace y by -y | Curve is symmetric about the X–axis |
| About the Y–axis | (x, y) ∈ C ⇔ (-x, y) ∈ C | Replace x by -x | Curve is symmetric about the Y–axis |
| About Origin | Equation unchanged when both signs change | Replace x → -x, y → -y | The curve is symmetric about Origin |
Key Points: Standard Curves
Parabola:
-
y2 = 4ax → opens right
-
y2 = −4ax → opens left
-
x2 = 4ay → opens upward
-
x2 = −4ay → opens downward
Ellipse:
-
\[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\quad(a>b)\]
-
\[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\quad(a<b)\]
Important Questions [9]
- Find the area of the region bounded by the parabola y2 = 25x and the line x = 5.
- Find the area of the region bounded by the parabola y^2 = 4x and the line x = 3.
- Area of the region bounded by y = x4, x = 1, x = 5 and the X-axis is _______.
- Find the area of the region bounded by y = x2, the X-axis and x = 1, x = 4.
- The slope of a tangent to the curve y = 3x2 – x + 1 at (1, 3) is ______.
- The area of the region bounded by the curve y = x2, x = 0, x = 3, and the X-axis is ______.
- Find the area between the two curves (parabolas) y2 = 7x and x2 = 7y.
- Find the Area of the Region Bounded by the Parabola Y2 = 16x and the Line X = 4.
- Find the area of the regions bounded by the line y = −2x, the X-axis and the lines x = −1 and x = 2.
