Equation of Tangent

i. Point Form

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Differentiate w.r.t. $\mathrm{x}$

$\begin{array}{rl}& \frac{2x}{a^2}-\frac{2y}{a_2}\frac{dy}{dx}=0\\ & \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{{\mathrm{b}}^2\mathrm{x}}{{\mathrm{b}}^2\mathrm{y}}\end{array}$

Slope of tangent $=\frac{b^2{x}_1}{a^2{y}_1}$

Equation of tangent, $y-y_1=\frac{b^2{x}_1}{a^2{y}_1}(x-x_1)$

$\frac{{\mathrm{xx}}_1}{{\mathrm{a}}^2}-\frac{{\mathrm{yy}}_1}{{\mathrm{b}}^2}=\frac{{\mathrm{x}}_1^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}_1^2}{{\mathrm{b}}^2}$

$\frac{{\mathrm{xx}}_1}{{\mathrm{a}}^2}-\frac{{\mathrm{yy}}_1}{{\mathrm{b}}^2}=1(\text{ But }\mathrm{P}({\mathrm{x}}_1,{\mathrm{y}}_1)$ lies on hyperbola$)$

or $\mathrm{T}=\frac{{\mathrm{xx}}_1}{{\mathrm{a}}^2}-\frac{{\mathrm{yy}}_1}{{\mathrm{b}}^2}-1$

Equation of tangent is $\mathrm{T}=0$

Equation of tangent at point $\mathrm{P}({\mathrm{x}}_1,{\mathrm{y}}_1)$ to the hyperbola $\frac{(\mathrm{x}-\mathrm{h}{)}^2}{{\mathrm{a}}^2}-\frac{(\mathrm{y}-\mathrm{k}{)}^2}{{\mathrm{b}}^2}=1$ is $\frac{(\mathrm{x}-\mathrm{h})({\mathrm{x}}_1-\mathrm{h})}{{\mathrm{a}}^2}-\frac{(\mathrm{y}-\mathrm{k})({\mathrm{y}}_1-\mathrm{k})}{{\mathrm{b}}^2}=1$

ii. Parametric Form

Parametric equation of hyperbola is $x=a\sec \theta ,y=b\tan \theta$

Equation of tangent is

$\frac{x}{a}\sec \theta -\frac{y}{b}\tan \theta =1$

iii. Slope Form

$y=mx\pm \sqrt{a^2{m}^2-b^2}$

Examples

1. The equation of a tangent to the hyperbola $16x^2-25y^2-96x+100y-356=0$, which makes an angle $\frac{\pi }{4}$ with the transverse axis, is

(a). $\mathrm{y}=\mathrm{x}+2$

(b). $\mathrm{y}=\mathrm{x}-5$

(c). $\mathrm{y}=\mathrm{x}+3$

(d). $x=y+2$

2. The point of intersection of two tangents to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the product of whose slopes is ${\mathrm{c}}^2$, lies on the curve

(a). ${\mathrm{y}}^2-{\mathrm{b}}^2={\mathrm{c}}^2({\mathrm{x}}^2+{\mathrm{a}}^2)$

(b). ${\mathrm{y}}^2+{\mathrm{a}}^2={\mathrm{c}}^2({\mathrm{x}}^2-{\mathrm{b}}^2)$

(c). ${\mathrm{y}}^2+{\mathrm{b}}^2={\mathrm{c}}^2({\mathrm{x}}^2-{\mathrm{a}}^2)$

(d). ${\mathrm{y}}^2-{\mathrm{a}}^2={\mathrm{c}}^2({\mathrm{x}}^2+{\mathrm{b}}^2)$

3. If the tangents drawn from a point on the hyperbola $x^2-y^2=a^2-b^2$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ make angles $\alpha$ and $\beta$ with the transverse axis of the hyperbola, then

(a). $\tan \alpha -\tan \beta =1$

(b). $\tan \alpha +\tan \beta =1$

(c). $\tan \alpha \cdot \tan \beta =1$

(d). $\tan \alpha \cdot \tan \beta =-1$

4. If the line $2x+\sqrt{6}y=2$ touches the hyperbola $x^2-2y^2=4$, then the point of contact is

(a). $(-2,\sqrt{6})$

(b). $(-5,2\sqrt{6})$

(c). $(\frac{1}{2},\frac{1}{\sqrt{6}})$

(d). $(4,-\sqrt{6})$

5. Number of real tangents can be drawn from the point $(5,0)$ to the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ is

(a). 2

(b). 1

(c). 0

(d). 4

Practice questions

1. $\mathrm{P}$ is a point on the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1,\mathrm{N}$ is the foot of the perpendicuilar from $\mathrm{P}$ on the transverse axis. The tangent to the hyperbola at $\mathrm{P}$ meets the transverse axis at $\mathrm{T}$. If $\mathrm{O}$ is the centre of the hyperbola, then $\mathrm{OT}\times \mathrm{ON}$ is

(a). $b^2$

(b). $a^2$

(c). $ab$

(d). none of these

2. Tangents drawn from the point (c,d) to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ make angles $\alpha$ and $\beta$ with the $\mathrm{x}$-axis. If $\tan \alpha \tan \beta =1$, then ${\mathrm{c}}^2-{\mathrm{d}}^2=$

(a). $a^2-b^2$

(b). $a^2\cdot b^2$

(c). $a^2+b^2$

(d). none of these

3. The common tangents to the two hyperbolas $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ and $\frac{{\mathrm{y}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{x}}^2}{{\mathrm{b}}^2}=1$ is

(a). $\mathrm{y}=\mathrm{x}+\sqrt{{\mathrm{a}}^2-{\mathrm{b}}^2}$

(b). $y=x+\sqrt{a^2+b^2}$

(c). $y=2x+\sqrt{a^2-b^2}$

(d). none of these

4. The equation of the tangent to the curve $4x^2-9y^2=1$ which is parallel to $4y=5x+7$ is

(a). $30x+24y=720$

(b). $7y=6x-15$

(c). $y=3\sqrt{\frac{2}{7}}x+\frac{15}{\sqrt{7}}$

(d). none of these

5. The locus of a point $\mathrm{P}(\mathrm{h},\mathrm{k})$ moving under the condition that the line $\mathrm{y}=\mathrm{hx}+\mathrm{k}$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is

(a). Parabola

(b). circle

(c). ellipse

(d). hyperbola