91Ó°ÊÓ

In engineering the eccentricity of an ellipse is often expressed only in terms of \(a\) and \(b\). Show that \(e=\sqrt{1-b^{2} / a^{2}}\)

The vertical position of a projectile is given by the equation \(y=-16 t^{2}\) and the horizontal position by \(x=40 t\) for \(t \geq 0 .\) By eliminating \(t\) between the two equations, show that the path of the projectile is a parabolic arc. Graph the path of the projectile.

The point on a parabola closest to the focus is the vertex. How would you go about proving this? Carry out your ideas.

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ \mathbf{u}+(\mathbf{v}+\mathbf{w}) $$

Tangent Lines In this problem it is necessary that you have studied Section 2.10 . (a) Any two distinct tangent lines to a parabola must intersect. Show that the tangent lines to a parabola at the endpoints of its focal chord are perpendicular. [Hint: Without loss of generality we may assume that an equation of the parabola is given by \(x^{2}=4 c y\).] (b) Show that the tangent lines at the endpoints of the focal chord of a parabola intersect on its directrix.

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ 2 \mathbf{u}-(\mathbf{v}-\mathbf{w}) $$

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ \mathbf{v}+2(\mathbf{u}-3 \mathbf{w}) $$

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ 4(\mathbf{u}+2 \mathbf{w})-6 \mathbf{v} $$

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ |\mathbf{u}+\mathbf{w}| $$

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ |\mathbf{w}||2 \mathbf{v}| $$