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 \(\frac{dy}{dx}\)](x_o.y_o)=\(\frac{b^2x_0}{a^2y_0}\).

Then, the equation of the tangent at (x_o,y_o) 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 (x_o,y_o) is given by,

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