Question

Find the equations of the tangent and normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}$=1at the point (x0,y0).

Answer 1

Find the equations of the tangent and normal to the hyperbola x2/a2-y2/b2=1 at the point (x0,y0).

Differentiating $\frac{x^2}{a^2}-\frac{y^2}{b^2}$=1 with respect to x, we have:

$\frac{2x}{a^2}-\frac{2y}{b^2}\frac{dy}{dx}=0$

$\frac{2y}{b^2}\frac{dy}{dx}=\frac{2x}{a^2}$

$\frac{dy}{dx}=\frac{b^2x}{a^2x}$

Therefore, the slope of the tangent at (x0,y0) is $\frac{dy}{dx}$](xo.yo)=$\frac{b^2x_0}{a^2y_0}$.

Then, the equation of the tangent at (xo,yo) is given by,

y-y0=$\frac{b^2x_0}{a^2yy_0}-a^2y^2_0=b^2xx_0-b^2x^2_0$

$b^2xx_0-a^2yy_0-b^2x^2_0+a^2y^2=0$

$\frac{xx_0}{a^2}-\frac{yy_0}{b^2}-1=0$

Hence, the equation of the normal at (xo,yo) is given by,

=$y-\frac{y_0}{a^2y_0}+\frac{(x-x)}{b^2x_0}=0$