determine the point on the hyperbola 6x2−5y2=106x2−5y2=10 closest to the point (0, -3).
Answers
The point on the hyperbola closest to (0, -3) is (√2, -3√2).
What is Distance formula ?The distance formula is a formula used to calculate the distance between two points in a coordinate plane or in a three-dimensional space.
For two points in a two-dimensional coordinate plane, \((x_1, y_1) and (x_2, y_2)\), the distance formula is:
d = \(\sqrt{[(x_2 - x_1)^2 + (y_2 - y_1)^2]}\)
To find the point on the hyperbola closest to the point (0, -3), we need to find the point on the hyperbola where the distance to (0, -3) is minimized.
We can use the distance formula to find the distance between any point on the hyperbola (x, y) and (0, -3):
d = √[(x - 0)² + (y - (-3))²]= √[x² + (y + 3)²]
We want to minimize this distance, subject to the constraint that (x, y) lies on the hyperbola 6x² - 5y² = 10.
To solve this problem, we can use Lagrange multipliers. Let L(x, y, λ) = d² - λ(6x² - 5y² - 10), where λ is the Lagrange multiplier. We square the distance to avoid dealing with the square root. We also subtract λ times the constraint function, since we want to maximize the distance subject to the constraint.
Taking partial derivatives with respect to x, y, and λ, we get:
∂L/∂x = 2x - 12λx = 0
∂L/∂y = 2(y + 3) + 10λy = 0
∂L/∂λ = 6x² - 5y² + 10 = 0
Solving the first equation for λ and substituting into the second equation, we get:
λ = 1/2x
2(y + 3) + 5yx = 0
Solving for y in terms of x, we get:
y = -6/(5x)
Substituting into the equation for the hyperbola, we get:
6x² - 5(-6)²/(5x)² = 10
Multiplying both sides by (5x)², we get:
\(30x^4 + 180 = 25x^2\)
Rearranging and factoring, we get:
30(x² - 2)(x² + 3) = 0
The only positive value of x that satisfies this equation is x = √2. Substituting into y = -6/(5x), we get y = -3√2. Therefore, the point on the hyperbola closest to (0, -3) is (√2, -3√2).
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Related Questions
Hewp:(
Using compatible numbers, which is the best estimate of (28.5)(7.1)?
36
210
300
Answers
Answer:
210
Step-by-step explanation:
Using compatible numbers, which is the best estimate of (28.5)(7.1)?
Answer: So let's round 28.5 to 29 because 4 and below let it rest. 7.1 is gonna be 7. Now, 29x7= 203 and since 300 and 36 are way off, we find the answer as 210(B)
Step-by-step explanation:
Please help thank you so much in advance
Answers
Answer:
Its 70º
Step-by-step explanation:
Find the coordinates of the centroid of the triangle with the given vertices.
F(1, 5), G(-2, 7), H( – 6, 3)
Answers
The coordinate of the centroid of the given triangle will be at (-2.33,5).
What is a triangle?A triangle is a 3-sided shape that is occasionally referred to as a triangle. There are three sides and three angles in every triangle, some of which may be the same.
The given triangle with vertices has been drawn.
The midpoint of H( – 6, 3) and F(1, 5) will be as,
x = (-6 + 1)/2 = -2.5
y = (3 + 5)/2 = 4 so D(-2.5,4)
The coordinate of the centroid will intersect 2:1 of the median from the vertex side.
Thus by intercept formula,
x = (2 × -2.5 + 1 × -2)/(2 + 1) and y = (2 × 4 + 1 × 7)/(2 + 1)
x = -2.33 and y = 5
So the coordinate of vertices will be (-2.33,5).
Hence "The specified triangle's centroid's coordinate will be at (-2.33,5)".
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Choose the correct description of the graph of the inequality X - 3 greater than or equal to 25
A. Open circle on 8, shading to the left
B. Closed circle on 8, shading to the left.
C. Open circle on 8, shading to the right.
D. Closed circle on 8, shading to the right.
I’m pretty sure it’s D
Answers
Answer:
D. Closed circle on 8, shading to the right.
Consider this function in recursive form.
A1 = -8
An = 3 + an-1
Enter an equivalent equation written in explicit form where n is the term number.
Type your response in simplest form. Do not type any spaces in your response.
an
An = _____
Answers
Answer:
Step-by-step explanation:
Io
Use the Pythagorean Theorem to find the diagonal.
a- 60
b- 84.85
c- 10.95
d- 7.74
Answers
Answer:
84.85
Step-by-step explanation:
\( {a}^{2} + {b}^{2} = {c}^{2} \)
\( {60}^{2} + {60}^{2} = {c}^{2} \\ 3600 + 3600 = {c}^{2} \\ 7200 = {c}^{2} \)
now we should find the square root of 7200 to find c
\( \sqrt[]{7200} = 85.85.....\)
find a positive integer n that has a last decimal digit 7 and is not in the set s from the previous problem. prove that n is not in s.
Answers
In the previous problem, we had to find a set of positive integers such that no number in the set has a last decimal digit of 7. Now we have to find a positive integer n that has a last decimal digit of 7 and is not in that set S. Let's say that S is the set of positive integers that do not have a last decimal digit of 7.
We can show that there is a positive integer n that has a last decimal digit of 7 and is not in S. Suppose that there is no such positive integer. Then every positive integer must either have a last decimal digit of 7 or be in S. But this would mean that the union of S and the set of positive integers with a last decimal digit of 7 would be the set of all positive integers, which is impossible. Therefore, there must be a positive integer n that has a last decimal digit of 7 and is not in S. To prove that n is not in S, we have to show that n has a last decimal digit of 7. If n were in S, it would not have a last decimal digit of 7. Therefore, n is not in S. In conclusion, we have found a positive integer n that has a last decimal digit of 7 and is not in S. This proves that S is not the set of all positive integers that do not have a last decimal digit of 7, since there is at least one positive integer that has a last decimal digit of 7 and is not in S.
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Graph the picture nes whose equation is 2y=-3x-2
Answers
Answer:
Picture Below
Step-by-step explanation:
You can simplify it by dividing it into
y = -1.5x-1
what is the sum 10 + (-10)
Answers
Answer:
0
Step-by-step explanation:
because
k c r
10 + (-10)
10 - 10=0
Answer:
☛☚um its 0
Step-by-step explanation:
Last year you mowed grass and shoveled snow for 12 houses. You earned $225 for each lawn you mowed
and $200 for each house you shoveled for a total of $2600. How many lawns did you mow? How many
driveways did you shovel?
Answers
1800 lawns were mowed. 800 driveways were shoveled.
What is equation?A formula that expresses the connection between two expressions on each side of a sign. Typically, it has a single variable and an equal sign. Like this: 2x - 4 Equals 2. In the above example, the variable x exists. The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions.
Given Data
Let mowed grass be x
Let shoveled snow be y
Last year you mowed grass and shoveled snow for 12 houses
x + y = 12
y = 12 -x
You earned $225 for each lawn you mowed and $200 for each house you shoveled for a total of $2600.
225x + 200y = 2600
Equating value of y
225x + 200(12-x) = 2600
225x + 2400 - 200x = 2600
225x - 200x = 2600 - 2400
25x = 200
x = 8
For value of y,
y= 12 -x
y = 12 - 8
y = 4
Lawn mowed = 225(8)
Lawn mowed = 1800
Shoveled snow = 200(4)
Shoveled snow = 800
1800 lawns were mowed. 800 driveways were shoveled.
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Triangle MNO is isosceles. Find the value of y and the measure of Angle O. Y=______
Angle O=_______degrees
Answers
Answer:
y = -5
o = 35 degrees
Step-by-step explanation:
If you replace y with -5, the statement 7y = 4y - 15 is correct. You get -35 = -35. Hope this helped!
witch of the following one step transformations of figure p could Laney have done to draw figure q.
Answers
The one step transformations of figure p is reflection over the line x = 1.
option D.
What is the reflection of a figure?A reflection is a mirror image of a shape or figure. An image will reflect through a line, known as the line of reflection.
A figure is said to reflect the other figure, and then every point in a figure is equidistant from each corresponding point in another figure.
So from the given figure P, the figure Q is obtained through the reflection of x axis, particular on x = 1.
When reflecting a figure vertically across x = 1, we essentially flip it over like a mirror reflection across this axis.
By doing so, all points in our original image transform into new points positioned symmetrically to this straight line relative to their previous location - e.g., equidistant but on opposite sides, as shown in figure Q.
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The particular solution of X' = (2 3)X + ( t ) is
(2 1) ( 1 )
Select the correct answer. a. (t/4 + 19/16)
(-t/2 + 7/8) b. (t/4 - 19/16) (-t/2 + 7/8 ) c. (t/4 + 1/8)
(-t/2 - 7/8)
Answers
The particular solution is: Xp(t) = (t/4 - 19/16; -t/2 + 7/8). The correct option is b.
The given system of linear differential equations can be written as:
X'(t) = AX + B(t),
where X'(t) is the derivative of X(t), A is the matrix (2 3; 2 1), and B(t) is the column vector (t; 1). To find the particular solution, we can apply the method of undetermined coefficients. We assume a particular solution of the form Xp(t) = (at + b; ct + d), where a, b, c, and d are constants to be determined.
Taking the derivative of Xp(t), we get Xp'(t) = (a; c). Now, we substitute Xp(t) and Xp'(t) into the given equation:
(a; c) = (2 3; 2 1) (at + b; ct + d) + (t; 1).
Multiplying the matrix and vector, we get:
(a; c) = (2(at + b) + 3(ct + d); 2(at + b) + 1(ct + d)) + (t; 1).
Equating the components, we get the following system of linear equations:
a = 2a + 2b + 3c + 3d + 1,
c = 2a + 2b + c + d + 0.
Solving this system, we find a = t/4 - 19/16, b = -t/2 + 7/8. Therefore, the particular solution Xp(t) is:
Xp(t) = (t/4 - 19/16; -t/2 + 7/8),
which corresponds to option b.
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Help with geometry on equations of circles. How would I find the value of CN?
Answers
The required measure of the tangent CN is 7.
As we know, tangents drawn from a point outside the circle have the same or equal length, following this,
CN = AN
AM = BM
CK = BK
The perimeter of triangle KMN is equal to 34.
CN +AN + AM + BM + CK + BK = 34
CN +CN+ AM + AM+ BK+ BK = 34
2[CN + AM + BK] = 34
CN + 6 + 4 = 17
CN = 17 - 10 = 7
Thus, the required measure of the tangent CN is 7.
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Water flows through a pipe at a rate of 8600 cubic meters per hour. Express this rate of flow in quarts per second. Round your answer to the nearest whole number.
Answers
Answer:
R = 2524 quarts/s
Step-by-step explanation:
Water flows through a pipe at a rate of 8600 cubic meters per hour.
We know that,
1 m³ = 1056.69 quarts
1 hour = 3600 s
Rate of flow of water = 8600 cubic meters per hour
\(R=8600\ \dfrac{m^3}{h}\\\\=8600\times \dfrac{1056.69\ \text{quarts}}{3600\ s}\\\\=2524.311\ \text{quarts/s}\)
or
R = 2524 quarts/s
So, the rate of flow of water is 2524 quarts/s.
Urgent!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
The first round of a racquetball tournament begins with 148 players. Every round, half of the players lose and are eliminated from the tournament leaving 74 players in the second round. The tournament continues in the same fashion until one player finally wins the tournament. Determine if the scenario describes an arithmetic or a geometric sequence then write the formula for the sequence described.
Answers
Answer:
Notice that 99 players need to be eliminated for there to be declared a winner. Notice also that every match eliminates exactly one person. Therefore, 99 matches are needed to eliminate 99 people and therefore declare a winner. The rest of the information is irrelevant. Therefore, the total number of matches is text divisible by 11.
Find the absolute value of -0.02
Answers
Explanation: the absolute value of anything is the distance between the given number and zero.
The owner of a video game store creates the expression −2x2 28x 20 to represent the store's weekly profit in dollars, where x represents the price of a new video game. choose the equivalent expression that reveals the video game price that produces the highest weekly profit, and use it to determine that price. −2(x − 7)2 118; x = $7 −2(x − 7)2 118; x = $118 −2(x2 − 14x) 20; x = $14 −2(x2 − 14x − 10); x = $10
Answers
The given expression is equivalent to \(\rm -2(x-7)^2+118\)price is $7 option first is correct.
What is a quadratic expression?A quadratic expression is a second-degree algebraic expression represented by the:
\(\rm ax^2+bx+c=0\) ........(1)
Where a, b, and c are the constants and
We have a quadratic expression:
\(\rm f(x)=-2x^2+28x+20\)
Convert it to perfect square by perfect square trinomial:
\(\rm f(x)=-2(x^2+14x)+20\)
\(\rm f(x)=-2(x^2+14x+49-49)+20\) (adding and subtracting by 49)
\(\rm f(x)=-2(x-7)^2+2(49)+20\)
\(\rm f(x)=-2(x-7)^2+118\)
Here \(\rm (x-7)^2\) is the positive term and -2 is the negative term so in order to get the highest weekly profit it should be zero, so
\(\rm -2(x-7)^2=0\)
x = $7
Thus, the given expression is equivalent to \(\rm -2(x-7)^2+118\) and the price is $7.
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Answer:
First option A. −2(x − 7)^2 + 118; x = $7
Step-by-step explanation:
Took the test and it's correct, trust me :)
Find the missing side length
I WILL GIVE BRAINLIEST TO CORRECT ANSWER
Answers
Triangle $ABC$ has its vertices $A$, $B$, and $C$ on the sides of a rectangle 4 units by 5 units as shown. What is the area of triangle $ABC$ in square units?
Answers
Answer:
10 square units.
Step-by-step explanation:
It is given that triangle ABC has its vertices A, B, and C on the sides of a rectangle 4 units by 5 units.
Height of triangle = 4 units
Base of triangle = 5 units
We need to find the area of triangle.
Area of triangle \(=\dfrac{1}{2}\times base \times height\)
Substitute base = 5 and height = 4 in the above formula.
Area of triangle \(=\dfrac{1}{2}\times 5\times 4\)
Area of triangle \(=\dfrac{20}{2}\)
Area of triangle \(=10\)
Therefore, the area of triangle ABC is 10 square units.
Answer:
9
Step-by-step explanation:
true or false: the quantity represented by θ is a function of time (i.e., is not constant).
Answers
Answer: the answer to this is true
BD and AC are chords that intersect at point Y. What is the length of line segment AY? 2 units 3 units 4 units 6 units
i need it now please
Answers
Answer:
4 units
Step-by-step explanation:
The complete question is
BD and AC are chords that intersect at point Y. A circle is shown. Chords B D and A C intersect at point Y. The length of B Y is 3, the length of Y D is 8, the length of A Y is x, and the length of Y C is 6. What is the length of line segment AY?
The picture of the question in the attached figure
we know that
The Intersecting Chord Theorem states that: When two chords intersect each other inside a circle,the products of their segments are equal.
do
In this problem
substitute the given values
solve for x
therefore
The length of segment AY is 4 units
7. The floor area of a passage is 2,4 m². The floor has to be tiled with square tiles of which the length of the sides is 200 mm. How many tiles do you need to cover the floor?
Answers
The number of tiles needed to cover the floor is 60 tiles.
Given Information,
The floor area of the passage = 2.4 m²
Length of a side of square tile = 200mm
Length of the side is in mm which needs to be converted in its standard form which is meters.
Therefore, the length of side = 200/1000 m
= 0.2 m
Area of the square tile = 0.2 x 0.2
= 0.04 m²
Let the number of tiles be x.
∴ Number of tiles = \(\frac{Area of passage}{Area of each tile}\)
x = 2.4/0.04
x = 60
∴ The number of tiles needed to cover the floor is 60 tiles.
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Giving brainlyist to the first person who answers correctly
Answers
Answer: is it A
Step-by-step explanation:
Odd numbers are whole numbers that cannot be divided exactly into pairs. Odd numbers, when divided by 2, leave a remainder of 1. 1, 3, 5, 7, 9, 11, 13, 15 … are sequential odd numbers. Odd numbers have the digits 1, 3, 5, 7, or 9 in their one's place.
A 6 character computer password is made up of 4 numbers followed by 2 letters. How many different passwords are possible?
Answers
Answer: 6,760,000
Step-by-Step Explanation:
For a recent art project, a teacher purchased 94 pieces of red construction paper and 80 What is the exact ratio of red construction paper to blue construction paper?
Answers
In the recent art project, the teacher purchased 94 pieces of red construction paper and 80 pieces of blue construction paper. To determine the exact ratio of red construction paper to blue construction paper, we divide the number of red construction paper pieces by the number of blue construction paper pieces.
The ratio of red construction paper to blue construction paper can be expressed as:94:80
However, we can simplify this ratio by dividing both numbers by their greatest common divisor. In this case, the greatest common divisor of 94 and 80 is 2.
Dividing 94 by 2 gives us 47, and dividing 80 by 2 gives us 40.
Therefore, the exact simplified ratio of red construction paper to blue construction paper is: 47:40
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Choco pies are cookies made from chocolate (c), sugar (s), and flour (. Chocolate costs $5 per pound, sugar costs
$3 per pound, and flour costs $2 per pound. You spend $50 on 18 pounds of food and buy twice as much flour as
sugar
Calvin wants to invest $12,000 in deposits. For tax purposes, he wants his total interest to be $600. He wants to put
$1000 more in a 2-year deposit than in a l-year deposit and invest the rest in a 3-year deposit. How much money
should he invest in each deposit?
Number of Years
Rate
2.0%
5.09
6.0%
1-year: $2200, 2-year: $3200, 3-year: $6600
1-year: $4000, 2-year: S5000, 3-year: $3000
1-year: S3200, 2-year: $4200, 3-year: $4600
1-year: $1000, 2-year: S2000, 3-year: $9000
Answers
Question :
justinapace7446
Yesterday
Mathematics
High School
+5 pts
Choco pies are cookies made from chocolate (c), sugar (s), and flour (. Chocolate costs $5 per pound, sugar costs
$3 per pound, and flour costs $2 per pound. You spend $50 on 18 pounds of food and buy twice as much flour as
sugar
Calvin wants to invest $12,000 in deposits. For tax purposes, he wants his total interest to be $600. He wants to put
$1000 more in a 2-year deposit than in a l-year deposit and invest the rest in a 3-year deposit. How much money
should he invest in each deposit?
Number of Years
Rate
2.0%
5.0%
6.0%
Answer:
1-year: $2200, 2-year: $3200, 3-year: $6600
Step-by-step explanation:
Assume y = Amount invested for 1 year.
y + 1000 = investment amount for 2 years
12000 - (y + y + 1000 ) = amount invested for 3 years
11000 - 2y = amount invested for 3 years
Total interest
Multiplying each by its corresponding rate
(0.02 × y) + (0.05 × (y+1000)) + (0.06 × (11000-2y)) = 600
0.02y + 0.05y + 50 + 660 - 0.12y = 600
-0.05y + 710 = 600
-0.05y = 600 - 710
-0.05y = - 110
y = 2200
Therefore first year = y = $2200
Investment for 2 years =y + 1000 = 2200 + 1000 = $3200
Investment for 3 years = 11000 - 2y = (11000 - (2*2200)) = (11000 - 4400) = $6600
What is the next fraction in this sequence?Simplify your answer.
1/2, 4/7, 9/14, 5/7,....
Answers
Answer:
Step-by-step explanation:
\(\displaystyle\\\frac{1}{2},\ \ \frac{4}{7},\ \ \frac{9}{14},\ \ \frac{5}{7} ,\ \ ... \\\\\frac{1*7}{2*7} ,\ \ \frac{4*2}{7*2} ,\ \ \frac{9}{14}, \ \ \frac{5*2}{7*2} ,\ \ ...\\\\\frac{7}{14},\ \ \frac{8}{14} , \ \ \frac{9}{14},\ \ \frac{10}{14},\ \ \frac{11}{14},\ \ ...,\ \ \frac{6+n}{14} , \ \ n=1,\ 2,\ 3,\ ...\)
How do I write 0.008 in word form? NEED HELP!!! ASAP...thanks!
Answers
Answer:
Step-by-step explanation:
eight thousandths" or "point zero zero eight" or "point oh oh eight." I don't really see the need to pronounce the zero to the left of the decimal point.
Someone please help me on #3 and 4 they are different problems. Please help me how to solve this step by step. (Sorry about the bad picture.)
Answers
Jack: six and a quarter = 6 1/4
Sister: three and a half = 3 1/2
6 + 3 = 9 + 1/2 = 9.5 + 1/4 = 9.75 cakes altogether
4.
She uses 10x the amount of flour, so will also use 10x the amount of eggs.
10 x 8 = 80 eggs