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lana66690 [7]
2 years ago
10

4. Find the standard from of the equation of a hyperbola whose foci are (-1,2), (5,2) and its vertices are end points of the dia

meter of the circle x² + y² - 4x - 4y +4= 0​
Mathematics
1 answer:
Ad libitum [116K]2 years ago
5 0

The <em>standard</em> form of the equation of the hyperbola that satisfies all conditions is (x - 2)²/4 - (y - 2)²/5 = 1 .

<h3>How to find the standard equation of a hyperbola</h3>

In this problem we must determine the equation of the hyperbola in its <em>standard</em> form from the coordinates of the foci and a <em>general</em> equation of a circle. Based on the location of the foci, we see that the axis of symmetry of the hyperbola is parallel to the x-axis. Besides, the center of the hyperbola is the midpoint of the line segment with the foci as endpoints:

(h, k) = 0.5 · (- 1, 2) + 0.5 · (5, 2)

(h, k) = (2, 2)

To determine whether it is possible that the vertices are endpoints of the diameter of the circle, we proceed to modify the <em>general</em> equation of the circle into its <em>standard</em> form.

If the vertices of the hyperbola are endpoints of the diameter of the circle, then the center of the circle must be the midpoint of the line segment. By algebra we find that:

x² + y² - 4 · x - 4 · y + 4= 0​

(x² - 4 · x + 4) + (y² - 4 · y + 4) = 4

(x - 2)² + (y - 2)² = 2²

The center of the circle is the midpoint of the line segment. Now we proceed to determine the vertices of the hyperbola:

V₁(x, y) = (0, 2), V₂(x, y) = (4, 2)

And the distance from the center to any of the vertices is 2 (<em>semi-major</em> distance, a) and the semi-minor distance is:

b = √(c² - a²)

b = √(3² - 2²)

b = √5

Therefore, the <em>standard</em> form of the equation of the hyperbola that satisfies all conditions is (x - 2)²/4 - (y - 2)²/5 = 1 .

To learn more on hyperbolae: brainly.com/question/27799190

#SPJ1

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In the figure below, segment CD is parallel to segment EF and point H bisects segment DE :
Grace [21]

Answer:

See explanation

Step-by-step explanation:

In the figure below, segment CD is parallel to segment EF, DE is a transversal, then angles DIH and HGI are congruent as alternate interior angles when two parallel lines are cut by a transversal.

Consider triangles DIH and EGH. In these triangles,

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Thus,

\triangle DIH\cong \triangle EGH by AAS postulate

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I need help with a math problem
soldi70 [24.7K]

Answer:

19,680

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Select all the correct answers. Which two inequalities can be used to find the solution to this absolute value inequality? 3|x +
ziro4ka [17]

The absolute value inequality can be decomposed into two simpler ones.

x < 0

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<h3></h3><h3>Which two inequalities can be used?</h3>

Here we start with the inequality:

3|x + 4| - 5 < 7

First we need to isolate the absolute value part:

3|x + 4| < 7 + 5

|x + 4| < (7 + 5)/3

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Solving both of these we get:

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Learn more about inequalities:

brainly.com/question/24372553

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Maya is going to an amusement park. The price of admission into the park is $10, and once she is inside the park she will have t
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Step by step directions Square root for 480
m_a_m_a [10]
<span>  <span>first off your answer is 21.90 and the step by step i  wrote it for you:) Finding the square root of a number is the inverse operation of squaring that number. Remember, the square of a number is that number times itself. </span> The perfect squares are the squares of the whole numbers. The square root of a number, n, written below is the number that gives n when multiplied by itself.   </span>                                                                                                                                                                          <span>Many mathematical operations have an inverse, or opposite, operation. Subtraction is the opposite of addition, division is the inverse of multiplication, and so on. Squaring, which we learned about in a previous lesson (exponents), has an inverse too, called "finding the square root." Remember, the square of a number is that number times itself. The perfect squares are the squares of the whole numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 … </span> The square root of a number, n, written <span> is the number that gives n when multiplied by itself. For example,</span> <span>because 10 x 10 = 100</span> Examples Here are the square roots of all the perfect squares from 1 to 100. Finding square roots of of numbers that aren't perfect squares without a calculator 1. Estimate - first, get as close as you can by finding two perfect square roots your number is between. 2. Divide - divide your number by one of those square roots.
3. Average - take the average of the result of step 2 and the root. <span>4. Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.
</span> Example: Calculate the square root of 10 () to 2 decimal places. <span>1. Find the two perfect square numbers it lies between.
</span> <span><span>Solution:
</span><span>32 = 9 and 42 = 16, so lies between 3 and 4.</span></span> <span>2. Divide 10 by 3. 10/3 = 3.33 (you can round off your answer)</span> <span>3. Average 3.33 and 3. (3.33 + 3)/2 = 3.1667</span> <span>Repeat step 2: 10/3.1667 = 3.1579</span><span>Repeat step 3: Average 3.1579 and 3.1667. (3.1579 + 3.1667)/2 = 3.1623</span> Try the answer --> Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001 If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3. <span>Note: There are a number of ways to calculate square roots without a calculator. This is only one of them.</span>         <span><span>
</span> </span>
<span>  <span />Example: Calculate the square root of 10 () to 2 decimal places. <span>1. Find the two perfect square numbers it lies between.
</span> <span><span>Solution:
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