91Ó°ÊÓ

Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$ y^{2}=20 x,(2,-2 \sqrt{10}) $$

Find the equation of the given central conic. Vertical hyperbola with eccentricity \(\sqrt{6} / 2\) that passes through \((2,4)\)

find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=3-2 \cos t, y=-1+5 \sin t ; t \neq n \pi $$

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{-4}{\cos \theta} $$

Sketch the graph of the given equation. \(x^{2}-4 y^{2}-14 x-32 y-11=0\)

Let \(r=f(\theta)\), where \(f\) is continuous on the closed interval \([\alpha, \beta] .\) Derive the following formula for the length \(L\) of the corresponding 26. polar curve from \(\theta=\alpha\) to \(\theta=\beta\). $$ L=\int_{\alpha}^{\beta} \sqrt{[f(\theta)]^{2}+\left[f^{\prime}(\theta)\right]^{2}} d \theta $$

Find the equation of the given central conic. Ellipse with foci \((\pm 2,0)\) and directrices \(x=\pm 8\)

The slope of the tangent line to the parabola \(y^{2}=5 x\) at a certain point on the parabola is \(\sqrt{5} / 4 .\) Find the coordinates of that point. Make a sketch.

Sketch the graph of the given equation. \(4 x^{2}+16 x-16 y+32=0\)

find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=3 \tan t-1, y=5 \sec t+2 ; t \neq \frac{(2 n+1) \pi}{2} $$