Conic Sections
Ellipse
Tips:
- A conic is said to be an ellipse if its eccentricity is less than 1 (0 < e < 1)
- A second degree non-homogenous equation ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0 represents an ellipse if h^{2} – ab < 0 and Δ ≠ 0
- Standard equation of the ellipse is \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}= 1 if a > b.
- If p, q are the segments of a focal chord of an ellipse b^{2}x^{2} + a^{2}y^{2} = a^{2}b^{2} , then condition is b^{2}(p + q) = 2a pq
- Equation of tangent to the ellipse \tt \frac{(x-\alpha^{2})}{a^{2}}+\frac{(y-\beta^{2})}{b^{2}}= 1 having slope ‘m’ is \tt y-\beta = m\left(x-\alpha\right)\pm\sqrt{a^{2}m^{2}+b^{2}}
- The midpoint of the chord lx + my + n = 0 of the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\ is\ \left(\frac{-a^{2}ln}{a^{2}l^{2}+b^{2}m^{2}},\frac{-b^{2}mn}{a^{2}l^{2}+b^{2}m^{2}}\right)
- The area of an ellipse S = 0 is π ab sq.units
- The maximum area of a rectangle that can be inscribed in the ellipse S = 0 is 2ab sq.units and the sides are \tt a\sqrt{2}, b\sqrt{2}
- The product of the perpendiculars drawn from the foci on any tangent of the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 is b^{2} if (a > b) is a^{2} (if a < b)
- The locus of the foot of the perpendicular drawn from the centre upon any tangent to the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 is (x^{2}+y^{2})^{2} = a^{2}x^{2} + b^{2}y^{2}
- Two tangents can be drawn to an ellipse from an external point.
- If y = mx + c is a tangent to the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 then the condition is c^{2} = a^{2}m^{2} + b^{2} and point of contact is \tt \left(-\frac{a^{2}m}{c},\frac{b^{2}}{c}\right)
- Equation of the auxiliary circle of the ellipse S = 0 is x^{2}+y^{2} = a^{2} (a > b) is x^{2} + y^{2} = b^{2} (a < b)
- Two straight lines y = m_{1}x and y = m_{2} x are conjugate diameters of the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,\ if\ m_{1}\times m_{2} = \frac{-b^{2}}{a^{2}}
- The eccentric angles of the ends of a pair of conjugate diameter of an ellipse differ by a right angle.
- The sum of the squares of the two conjugate semi diameters of an ellipse is contact and is equal to the sum of the squares of the semi axes of the ellipse. (i.e a^{2}+b^{2})
- Circle with focal length as diameter touches auxiliary circle.
- Perpendicular from focus to any tangent and line join of centre, point of contact intersect the corresponding directrix.
Introduction to Ellipse
Equation of Ellipse
Latus Rectum of Ellipse
Some Examples to Ellipse
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1. The equation of ellipse in standard form will be given by \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
2. If the centre of the ellipse is at point (h, k) and the directions of the axes are parallel to the coordinate axes , then its equation is \tt \frac{\left(x-h\right)^{2}}{a^{2}}+\frac{\left(y-k\right)^{2}}{b^{2}}=1.
3.
Imp. terms \ Ellipse | \tt \left\{\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\right\} | |
For a > b | For b > a | |
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) |
Equation of directrices | x = ± a / e | y = ± b / e |
Relation in a, b and e | b^{2} = a^{2} (1 − e^{2}) | a^{2} = b^{2} (1 − e^{2}) |
Length of latus-rectum | \tt \frac{2b^{2}}{a} | \tt \frac{2a^{2}}{b} |
Ends of latus - rectum | \tt \left(\pm ae, \pm \frac{b^{2}}{a}\right) | \tt \left(\pm \frac{a^{2}}{b},\pm be\right) |
Parametric equations | (a cos Φ, b sin Φ) | (a cos Φ, b sin Φ) (0 ≤ Φ < 2π) |
Focal radii | SP = a − ex_{1} SP = a + ex_{1} |
SP = b − ey_{1} SP = b + ey_{1} |
Sum of focal radii SP + S'P = | 2a | 2b |
Distance between foci | 2ae | 2be |
Distance between directrices | 2a/e | 2b/e |
Tangents at the vertices | x = −a, x = a | y = b, y = −b |
4. Equation of Chord
Let P (a cos θ, b sin θ) and Q (a cos φ, b sin φ) be any two points of the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1.
a). The equation of the chord joining these points will be \tt \left(y-b\sin \theta\right) = \frac{b \sin \phi - b \sin \theta}{a cos \phi - a \sin \theta}\left(x-a \ cos \theta\right)
or \tt \frac{x}{a}\cos \left(\frac{\theta + \phi}{2}\right)+\frac{y}{b}\sin \left(\frac{\theta + \phi}{2}\right)=\cos \left(\frac{\theta - \phi}{2}\right)
b). The equation of the chord of contact fo tangents drawn from a point (x_{1}, y_{1}) to the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ is \ \frac{xx_{1}}{a^{2}}+\frac{yy_{1}}{b^{2}}=1
c). The equation of the chord of the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 bisected at the point (x_{1}, y_{1}) is given by \tt \frac{xx_{1}}{a^{2}}+\frac{yy_{1}}{b^{2}}-1=\frac{x_1^2}{a^{2}}+\frac{y_1^2}{b^{2}}-1
5. Point Form The equation of the tangent to the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 at the point (x_{1}, y_{1}) is \tt \frac{xx_{1}}{a^{2}}+\frac{yy_{1}}{b^{2}}=1.
b). Parametric Form The equation of the tangent to the ellipse at the point (a cos θ, b sin θ) is \tt \frac{x}{a}\cos \theta+\frac{y}{b}\sin \theta=1.
c). Slope Form The equation of the tangent of slope m to the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ are \ y =mx\pm\sqrt{a^{2}m^{2}+b^{2}} and the coordinates of the point of contact are \tt \left(\pm \frac{a^{2}m}{\sqrt{a^{2}m^{2}+b^{2}}},\mp\frac{b^{2}m}{\sqrt{a^{2}m^{2}+b^{2}}}\right).
6. Equation of Normal
a). Point Form The equation of the normal at (x_{1}, y_{1}) to the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 is \tt \frac{a^{2}x}{x_{1}}-\frac{b^{2}y}{y_{1}}=a^{2}-b^{2}
b). Parametric Form The equation of the normal to the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a cos θ, b sin θ) is ax sec θ − by cosec θ = a^{2} − b^{2}
7. Slope Form The equation of the normal of slope m to the ellipse \tt \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 are given by \tt y = mx - \frac{m\left(a^{2}-b^{2}\right)}{\sqrt{a^{2}+b^{2}m^{2}}} and the coordinates of the point of contact are \tt \left(\pm \frac{a^{2}}{\sqrt{a^{2}+b^{2}m^{2}}},\pm \frac{b^{2}m}{\sqrt{a^{2}+b^{2}m^{2}}}\right)