• ### Nearest point on a hyperbola?

Hyperbola: 2x² - 7y² + 20 = 0 Point: (5, 0) Find the point(s) on the hyperbola nearest the given point.
Hyperbola: 2x² - 7y² + 20 = 0 Point: (5, 0) Find the point(s) on the hyperbola nearest the given point.
2 answers · Mathematics · 1 year ago
• ### Simplify the algebraic expression.?

√(9xy²z³) Few people seem to get these. On similar problems incorrect answers tend to prevail.
√(9xy²z³) Few people seem to get these. On similar problems incorrect answers tend to prevail.
5 answers · Mathematics · 3 years ago
• ### General term of a sequence?

These are the first four terms of a sequence: 1, 10, 28, 82 Find a general nth term that fits the sequence.
These are the first four terms of a sequence: 1, 10, 28, 82 Find a general nth term that fits the sequence.
5 answers · Mathematics · 5 years ago
• ### Construct a triangle quadrisection?

Given any arbitrary triangle, dissect it into four triangles of equal area, using in a pattern like that shown here. Three of them share a side with the given triangle, and the remaining one does not touch any of the given sides. This must be a compass and straightedge construction. Please do not offer answers with measurements or approximation... show more
Given any arbitrary triangle, dissect it into four triangles of equal area, using in a pattern like that shown here. Three of them share a side with the given triangle, and the remaining one does not touch any of the given sides. This must be a compass and straightedge construction. Please do not offer answers with measurements or approximation algorithms.
1 answer · Mathematics · 5 years ago
• ### Family of parabolas through three points?

I saw this question about a week ago. We were asked to derive the equation for the parabola through these three points in the Cartesian plane: (1, 11), (0, 6), (2, 18) My problem with the question is that there is more than one parabola fitting those three points. In fact, there are infinitely many. Everyone else was assuming a vertical axis,... show more
I saw this question about a week ago. We were asked to derive the equation for the parabola through these three points in the Cartesian plane: (1, 11), (0, 6), (2, 18) My problem with the question is that there is more than one parabola fitting those three points. In fact, there are infinitely many. Everyone else was assuming a vertical axis, which is probably what the asker intended, but that condition was not stated. So what about the others? Using a single variable parameter, derive an equation representing the family of parabolas passing through the three given points. Please read it carefully. The objective is not a single parabola, but rather a family of parabolas. I asked this same question two days ago, but was compelled to delete it because nobody was addressing the question.
Mathematics · 7 years ago
• ### Focus reflections in a conic?

Point F is a focus of an ellipse. Point P is the image of F when it is reflected on a line that is tangent to the ellipse. What is the locus of P as F is reflected on all tangent lines? Give a detailed description please. Answer the same question again for the cases of a parabola and a hyperbola.
Point F is a focus of an ellipse. Point P is the image of F when it is reflected on a line that is tangent to the ellipse. What is the locus of P as F is reflected on all tangent lines? Give a detailed description please. Answer the same question again for the cases of a parabola and a hyperbola.
1 answer · Mathematics · 8 years ago
• ### Chords bisected by a point?

Curve S and point P are defined below. S: 21x² + y² - 168x + 40y - 734 = 0 P(7, -29) How many chords of S are bisected by point P? Derive the equations of those chords.
Curve S and point P are defined below. S: 21x² + y² - 168x + 40y - 734 = 0 P(7, -29) How many chords of S are bisected by point P? Derive the equations of those chords.
3 answers · Mathematics · 8 years ago
• ### Locus of centers of orthogonal spheres?

Two spheres are given: x² + y² + z² + 14x − 4y − 147 = 0 x² + y² + z² + 4x − 28y - 168z + 6631 = 0 Write a single equation representing the locus of centers of spheres that are orthogonal to both of the given spheres.
Two spheres are given: x² + y² + z² + 14x − 4y − 147 = 0 x² + y² + z² + 4x − 28y - 168z + 6631 = 0 Write a single equation representing the locus of centers of spheres that are orthogonal to both of the given spheres.
2 answers · Mathematics · 8 years ago
• ### Can you keep this proof on the students' level?

This is something that came from an exercise book. Some students got stuck on this advanced problem. I can do the proof, but for sake of the students, I would like to find a simpler and shorter way. They have strong algebra skills, and they are getting good with elementary trigonometry identities, but they have had no double angle formulas and no... show more
This is something that came from an exercise book. Some students got stuck on this advanced problem. I can do the proof, but for sake of the students, I would like to find a simpler and shorter way. They have strong algebra skills, and they are getting good with elementary trigonometry identities, but they have had no double angle formulas and no calculus at all. This is what they have proved so far: sinθcosθ = k (sinθ + cosθ)² = 1 + 2k (sinθ - cosθ)² = 1 - 2k Now prove this inequality: -1/2 ≤ k ≤ 1/2
2 answers · Mathematics · 8 years ago
• ### What is the radius of these spheres?

Begin with one red sphere of unit radius. On its surface evenly distribute twenty congruent blue spheres. Each of the blue spheres is externally tangent to the red sphere and to exactly three of the other blue spheres. What is the radius of a blue sphere?
Begin with one red sphere of unit radius. On its surface evenly distribute twenty congruent blue spheres. Each of the blue spheres is externally tangent to the red sphere and to exactly three of the other blue spheres. What is the radius of a blue sphere?
1 answer · Mathematics · 8 years ago
• ### More reflections in an ellipse?

This is a follow-up to a question that was answered quite well a few days ago: http://answers.yahoo.com/question/index;_ylt=AtjuxUmS01GvMiHSJA2Coprsy6IX;_ylv=3?qid=20110511225722AAN4OoO From a point on an ellipse, a point particle is projected across the interior, tracing a chord. At the point where it intersects the ellipse it rebounds back... show more
This is a follow-up to a question that was answered quite well a few days ago: http://answers.yahoo.com/question/index;... From a point on an ellipse, a point particle is projected across the interior, tracing a chord. At the point where it intersects the ellipse it rebounds back across the interior again, subject to the reflective properties of an ellipse. It then traces another chord and rebounds again. This continues indefinitely. The first chord does not go through either focus. The path may or may not retrace the first chord. Suppose that it does not. Describe the pattern traced by the path.
1 answer · Mathematics · 8 years ago
• ### Locus of solutions to an equation?

Describe in detail the locus of points in the x-y plane satisfying this equation: (x² + y² - 14x + 8y + 65)(7x² + 16xy - 15y² - 7x + 5y) = 0
Describe in detail the locus of points in the x-y plane satisfying this equation: (x² + y² - 14x + 8y + 65)(7x² + 16xy - 15y² - 7x + 5y) = 0
3 answers · Mathematics · 8 years ago
• ### Reflections in an ellipse?

A ray is projected from a focus of an ellipse and reflected at the point where it intersects the ellipse. The path of the reflection goes through the other focus. This reflective property applies to all ellipses. But what happens after that? Continuing through the focus and onto the ellipse, it is reflected again and returns to the first focus, and... show more
A ray is projected from a focus of an ellipse and reflected at the point where it intersects the ellipse. The path of the reflection goes through the other focus. This reflective property applies to all ellipses. But what happens after that? Continuing through the focus and onto the ellipse, it is reflected again and returns to the first focus, and so on. In one trivial case, the first ray is coincident with the major axis, and the path is restricted to that axis. Are there any other cases in which the path would retrace itself? Does the path approach a stable orbit?
1 answer · Mathematics · 8 years ago
• ### One saddle, one minimum, and one maximum?

I received this from my nephew recently. I suppose we can forgo the sketching component. Can a differentiable function f(x,y) of two variables have on the plane exactly three critical points, one saddle, one local minimum, and one local maximum? If no, explain why, if yes, give an example and sketch a graph.
I received this from my nephew recently. I suppose we can forgo the sketching component. Can a differentiable function f(x,y) of two variables have on the plane exactly three critical points, one saddle, one local minimum, and one local maximum? If no, explain why, if yes, give an example and sketch a graph.
1 answer · Mathematics · 8 years ago
• ### Are these experiments equivalent?

A mathematics teacher, wishing to get caught up on some paperwork, gives his students a tedious activity of questionable educational merit. The students are paired off. Each team is given a fair cubic gaming die. They are instructed to roll the die 500 times and record the number of sixes. Looking up from his work, the teacher notices that one... show more
A mathematics teacher, wishing to get caught up on some paperwork, gives his students a tedious activity of questionable educational merit. The students are paired off. Each team is given a fair cubic gaming die. They are instructed to roll the die 500 times and record the number of sixes. Looking up from his work, the teacher notices that one team is not following his instructions to the letter. One student is rolling the die on a glass-top table and counting the sixes. The other student is sitting under the table and counting the sixes that appear on the bottom. The students insist that their procedure is equivalent to the one that was assigned. The probability of a six on bottom is equal to the probability of a six on top. This way, they argue, they can roll the die only 250 times and still record 500 trials. Are the students in fact conducting an equivalent experiment? Let X be the number of sixes recorded the usual way in 500 rolls. Let Y be the number of sixes recorded in 250 rolls using the modified procedure. Do X and Y have the same distribution?
2 answers · Mathematics · 8 years ago
• ### Locus of a circle center?

I put this up a couple of weeks ago, but received no correct answers. Can we try again? Two fixed, intersecting circles have unequal radii. A variable circle is tangent to both of the fixed circles. Describe the locus of the center of the variable circle.
I put this up a couple of weeks ago, but received no correct answers. Can we try again? Two fixed, intersecting circles have unequal radii. A variable circle is tangent to both of the fixed circles. Describe the locus of the center of the variable circle.
3 answers · Mathematics · 8 years ago
• ### The birthday problem revisited?

This concerns the classic birthday problem. Supposing that leap-day birthdays are not possible, 23 people are asked their birthdays. The probability that at least two of them match is greater than 1/2. Now I cannot recall the source, but long ago I read an account (supposedly historical) in which a mathematician was demonstrating it as a parlor... show more
This concerns the classic birthday problem. Supposing that leap-day birthdays are not possible, 23 people are asked their birthdays. The probability that at least two of them match is greater than 1/2. Now I cannot recall the source, but long ago I read an account (supposedly historical) in which a mathematician was demonstrating it as a parlor trick. After 22 party guests had given their birthdays, there were no matches. According to the source, the mathematician was still confident that there would be a match, because the probability favored it. Sure enough, the last guest shared a birthday with one of the others. Explain why the mathematician had no reason to be confident in success after there were no matches among the first 22. After how many birthdays with no match would the mathematician lose the advantage? That is, at what point would he first be in a position in which his probability of winning would be less than 1/2?
3 answers · Mathematics · 8 years ago
• ### Define this saw blade function?

How would you define this cyclic function? Its domain is all real numbers. Its graph is a series of congruent line segments forming a jagged edge, like a saw blade. Here are the vertices for three consecutive cycles: (-2, 0), (-1, 1), (0, 0), (1, 1), (2, 0), (3, 1), (4, 0) Connect the dots. At every differentiable point on the curve, the first... show more
How would you define this cyclic function? Its domain is all real numbers. Its graph is a series of congruent line segments forming a jagged edge, like a saw blade. Here are the vertices for three consecutive cycles: (-2, 0), (-1, 1), (0, 0), (1, 1), (2, 0), (3, 1), (4, 0) Connect the dots. At every differentiable point on the curve, the first derivative is either 1 or -1. There are many equivalent ways of defining the function. I am looking for the most concise definition formula. And it must be continuous on all real numbers, so leave no holes.
4 answers · Mathematics · 8 years ago
• ### Describe this geodesic on a cube?

A given cube has edges of unit length. A curve begins at the midpoint of an edge. From there it goes in a straight line across one face to the midpoint of an adjacent edge. It then continues in a geodesic path until it closes on itself at the point of beginning. What is the shape and the length of the curve?
A given cube has edges of unit length. A curve begins at the midpoint of an edge. From there it goes in a straight line across one face to the midpoint of an adjacent edge. It then continues in a geodesic path until it closes on itself at the point of beginning. What is the shape and the length of the curve?
1 answer · Mathematics · 8 years ago
• ### Ellipse construction problem?

Of course it is not possible to construct an ellipse with compass and straightedge, but what about this? Suppose that you are given an ellipse. Using compass and straightedge only, how could you construct the two axes? The usual construction rules apply (no measurements). It addition to line and arc intersections, you may use intersections with... show more
Of course it is not possible to construct an ellipse with compass and straightedge, but what about this? Suppose that you are given an ellipse. Using compass and straightedge only, how could you construct the two axes? The usual construction rules apply (no measurements). It addition to line and arc intersections, you may use intersections with the ellipse itself. You may assume that we are familiar with all of the elementary constructions such as angle bisection and parallel lines, so those steps may be mentioned without elaborating.
1 answer · Mathematics · 8 years ago