Tag: Incomplete

Prove that conics are the integral curves of a differential equation

by RoRi

May 31, 2016

Using the Cartesian equation for conics of eccentricity $e$ and center $(0,0)$ to prove that these conics are the integral curves of the differential equation

$y' = (e^2 - 1) \frac{x}{y}.$

Proof. Incomplete.

Prove that similar ellipses have the same eccentricity

by RoRi

May 31, 2016

Prove that a similarity transform carries an ellipse with center at the origin to another ellipse with the same eccentricity.
Prove the converse of part (a), i.e., two concentric ellipses with the same eccentricity and major axes on the same line are related by a similarity transform.
Prove the statements corresponding to parts (a) and (b) for hyperbolas.

Incomplete.

Prove that the points on each branch of a hyperbola satisfy a property

by RoRi

May 31, 2016

Prove that on each branch of a hyperbola the difference

$\lVert X - F \rVert - \lVert X + F \rVert$

is a constant.

Proof. Incomplete.

Prove that the sum of distances from a point on a ellipse to its foci is constant

by RoRi

May 31, 2016

Any conic section symmetric about the origin satisfies the equation

$\lVert X - F \rVert = | eX \cdot N - a| \qquad \text{where} \qquad a= ed + eF \cdot N.$

Using this, prove that if the conic section is an ellipse then we have

$\lVert X - F \rVert + \lVert X + F \rVert = 2a.$

This can be interpreted to say that the sum of the distances from a point on an ellipse to the foci is a constant.

Proof. Incomplete.

Compute some properties of ellipses and hyperbolas

by RoRi

May 31, 2016

Let $p$ be a positive number. Then the equation $px^2 + (p+2)y^2 = p^2 + 2p$ is the equation of an ellipse. Find (in terms of $p$ ) the eccentricity and the foci of this ellipse.
Find a Cartesian equation for the hyperbola with the same foci as in part (a) and which has eccentricity $e = \sqrt{3}$ .

Incomplete.

Find a Cartesian equation for a hyperbola through the origin

by RoRi

May 31, 2016

Find a Cartesian equation for the hyperbola which has asymptotes

$y = 2x + 1 \qquad \text{and} \qquad y = -2x + 3$

and which passes through the origin.

Incomplete.

Find the Cartesian equation for a parabola with given focus and directrix

by RoRi

May 31, 2016

Find a Cartesian equation for the parabola which has directrix $x + y + 1 = 0$ and whose focus is at the origin.

Incomplete.

Find the Cartesian equation for a conic section with given points

by RoRi

May 31, 2016

Find a Cartesian equation for a conic section which consists of the points $(x,y)$ such that the distance between $(x,y)$ and the point $(0,2)$ is half the distance from the point $(x,y)$ to the line $y = 8$ .

Incomplete.

Compute the area of a region enclosed by two parabolas

by RoRi

May 31, 2016

Consider the two parabolas with equations

$y^2 = 2(x-1), \qquad \text{and} \qquad y^2 = 4(x-2).$

These two parabolas enclose a region $R$ .

Compute the area of the region $R$ using integration.
Find the volume of the solid of revolution obtained by revolving $R$ about the $x$ -axis.
Find the volume of the solid of revolution obtained by revolving $R$ about the $y$ -axis.

Incomplete.

Find the volume of the solid of revolution obtained from a parabola

by RoRi

May 31, 2016

Find the volume of the solid of revolution generated by revolving the region bounded by the parabola $y = 8x^2$ and the line $x = 2$ about the $x$ -axis.

Incomplete.