Conic Sections
Hyperbola
- If the centre of the hyperbola lies at a point (h, k) and the axes are parallel to the co-ordinate axes, then the equation of the hyperbola is \tt \frac{\left(x-h\right)^{2}}{a^{2}}-\frac{\left(y-k\right)^{2}}{b^{2}}=1
- The condition for the line y = mx + c to be a tangent to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 is that c2 = a2m2 – b2 and the co-ordinates of the points of contact are \tt \left(\pm\frac{a^{2}m}{\sqrt{a^{2}m^{2}-b^{2}}},\pm\frac{b^{2}}{\sqrt{a^{2}m^{2}-b^{2}}}\right)
- The equation of tangent to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 interms of slope ‘m’ is \tt y = mx\ \pm\sqrt{a^{2}m^{2}-b^{2}}
- The equation of the normal to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 at the point (x1 y1) is \tt \frac{a^{2}x}{x_{1}}+\frac{b^{2}y}{y_{1}}=a^{2}+b^{2}
- The equation of the normal to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 at the point (a sec θ, b tan θ) is \tt \frac{ax}{\sec\theta}+\frac{by}{\tan\theta}=a^{2}+b^{2}
- The equation of normal to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 interms of slope ‘m’ is \tt y=mx\ \pm\ \frac{m\left(a^{2}+b^{2}\right)}{\sqrt{a^{2}-b^{2}m^{2}}}
- Equation of the pair of tangents drawn from a point P(x1 y1) to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 is SS1 = T2 where \tt S=\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}-1, S_{1}=\frac{x_{1}^{2}}{a^{2}}-\frac{y_{1}^{2}}{b^{2}}-1\ and\ T = \frac{xx_{1}}{a^{2}}-\frac{yy_{1}}{b^{2}}-1
- The equation of the chord of the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 with P(x1 y1) as its middle point is given by T = S1
- The equation of chord of contact of tangents drawn from a point P(x1 y1) to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 is T = 0
- Eccentricity of the hyperbola \tt (e) = \sqrt{\frac{a^{2}+b^{2}}{a^{2}}}
Tricks:
- Two tangents can be drawn from a point to a hyperbola. The two tangents are real and distinct or coincident or imaginary according as the given point lies outside, on or inside the hyperbola.
- The equation of director circle of the hyperbola is x2+y2 = a2 – b2
Equation of Hyperbola
Latus Rectum and Examples of Hyperbola
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1. If the centre of hyperbola is (h, k) and axes are parallel to the co-ordinate axes, then its equation is \tt \frac{\left(x-h\right)^{2}}{a^{2}}-\frac{\left(y-k\right)^{2}}{b^{2}}=1.
2.
| Imp. terms \ Hyperbola | \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 | \tt -\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ or \ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1 |
| Centre | (0, 0) | (0, 0) |
| Length of transverse axis | 2a | 2b |
| Length of conjugate axis | 2b | 2a |
| Foci | (± ae, 0) | (0, ± be) |
| Equation of directrices | x = ± a / e | y = ± b / e |
| Eccentricity | \tt e=\sqrt{\left(\frac{a^{2}+b^{2}}{a^{2}}\right)} | \tt e=\sqrt{\left(\frac{a^{2}+b^{2}}{b^{2}}\right)} |
| Length of latus rectum | 2b2 / a | 2a2 / b |
| Parametric co-ordinates | (a sec Φ, b tan Φ) 0 ≤ Φ < 2π |
(b sec Φ, a tan Φ) 0 ≤ Φ < 2π |
| Focal radii | SP = ex1 − a S'P = ex1 + a |
SP = ey1 − b S'P = ey1 + b |
| Difference of focal radii (S'P - SP) | 2a | 2b |
| Tangents at the vertices | x = − a, x = a | y = − b, y = b |
| Equation of the transverse axis | y = 0 | x = 0 |
| Equation of the conjugate axis | x = 0 | y = 0 |
3. Equations of chord joining two points P(a sec θ1, b tan θ1) and Q (a sec θ2 , b tan θ2) on the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1 is
\tt y - b tan \theta = \frac{b \tan \theta_{2}-b \tan\theta _{1}}{a \sec \theta_{2}-a \sec\theta _{1}}.\left(x-a \sec \theta_{1}\right)
or \frac{x}{a}\cos\left(\frac{\theta_{1}-\theta_{2}}{2}\right)-\frac{y}{b}\sin\left(\frac{\theta_{1}+\theta_{2}}{2}\right)=\cos\left(\frac{\theta_{1}+\theta_{2}}{2}\right)
4. Equation of chord of contact of tangents drawn from a point (x1, y1) to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \ is \frac{xx_{1}}{a^{2}}-\frac{yy_{1}}{b^{2}}=1.
5. The equation of the chord of the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 bisected at point (x1, y1) 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}}or \ T=S_{1}
Equation of tangent Hyperbola
a). Point Form The equation of the tangent to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \ at \ \left(x_{1},y_{1}\right)is \ \frac{xx_{1}}{a^{2}}-\frac{yy_{1}}{b^{2}}=1.
b). Parametric Form The equation of the tangent to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \ at \ \left(a\sec \theta,b \tan \theta\right)is \ \frac{x}{a}\sec \theta -\frac{y}{b}\tan \theta =1.
c). Slope Form The equation of the tangents of slope m to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 are given by \tt y = mx\pm \sqrt{a^{2}m^{2}-b^{2}}.
The coordinates of the point of contact are
\tt \left(\pm\frac{a^{2}m}{\sqrt{a^{2}m^{2}-b^{2}}},\pm\frac{a^{2}}{\sqrt{a^{2}m^{2}-b^{2}}}\right).
Normal Equation of Hyperbola
a). Point Form The equation of the normal to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \ is \frac{a^{2}x}{x_{1}}+\frac{b^{2}x}{y_{1}}= a^{2}+b^{2}.
b). Parametric Form The equation of the normal at (a sec θ, b tan θ) to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 is ax cos θ + by cot θ = a2 + b2.
c). Slope Form The equations of the normal of slope m to the hyperbola \tt \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 are given by \tt y=mx\mp\frac{m\left(a^{2}+b^{2}\right)}{\sqrt{a^{2}-b^{2}m^{2}}}