1:prove that the tangent at the point (1,-2) to the circle x²+y²=5 also touches the circle x²+y²-8x+6y+20=0 .Also find the point of contact.
2:find the length of tangent to the circle
3x²+3y²-7x-6y=12 drawn from (6,-7)​

Answers

Answer 1
#1

Circle 1 has center at (0, 0)

The tangent passes through the center and point (1, -2).

Equation of tangent at (1, -2):

x*1 + y*(-2) = 5x - 2y = 5x = 2y + 5

If the same line is tangent to circle 2 then they have one common point.

Solve the system:

x = 2y + 5x² + y² - 8x + 6y + 20 = 0

Substitute x:

(2y + 5)² + y² - 8(2y + 5) + 6y + 20 = 04y² + 20y + 25 - 16y - 40 + 6y + 20 = 05y² + 10y + 5 = 0y² + 2y + 1 = 0(y + 1)² = 0y + 1 = 0y = -1 ⇒ x = 2(-1) + 5 = -2 + 5 = 3

We see only one point of contact, (3, -1), this proves the line is tangent to circle 2 as well.

#2

Given circle:

3x² + 3y² - 7x - 6y = 12 and external point (6, -7).

Rewrite the equation of the circle as:

3x² + 3y² - 7x - 6y = 12x² + y² - 7/3x - 2y = 4x² - 2*7/6x + 49/36 + y² - 2y + 1 = 4 + 49/36 + 1(x - 7/6)² + (y - 1)² = 229/36

The center is (7/6, 1) the radius is √229/36

The length of the tangent t is the leg of the right triangle with another leg r being the radius and the hypotenuse d being the distance between the center and the external point:

t² = d² - r²t² = (6 - 7/6)² + (-7 - 1)² - 229/36t² = 36 - 14 + 49/36 + 64 - 229/36t² = 86 - 180/36t² = 81t = 9


Related Questions


Xavier shoots a basketball in which the height, in feet, is modeled by the equation,h(t) = -4t2 + 10 + 18, where t is time, in
seconds. What is the maximum height of the basketball?

Answers

Answer:

The maximum height of the basketball is of 24.25 feet.

Step-by-step explanation:

Vertex of a quadratic function:

Suppose we have a quadratic function in the following format:

[tex]f(x) = ax^{2} + bx + c[/tex]

It's vertex is the point [tex](x_{v}, y_{v})[/tex]

In which

[tex]x_{v} = -\frac{b}{2a}[/tex]

[tex]y_{v} = -\frac{\Delta}{4a}[/tex]

Where

[tex]\Delta = b^2-4ac[/tex]

If a<0, the vertex is a maximum point, that is, the maximum value happens at [tex]x_{v}[/tex], and it's value is [tex]y_{v}[/tex].

Height of the basketball:

Given by the following function:

[tex]h(t) = -4t^2 + 10t + 18[/tex]

Which is a quadratic function with [tex]a = -4, b = 10, c = 18[/tex]

What is the maximum height of the basketball?

y(in this case h) of the vertex. So

[tex]\Delta = b^2-4ac = 10^2 - 4(-4)(18) = 388[/tex]

[tex]y_{v} = -\frac{388}{4(-4)} = 24.25[/tex]

The maximum height of the basketball is of 24.25 feet.

Which is the better value for money 250g of coffee R12,35 or 450g of the same coffee at R21,95

Answers

Answer:

450g coffee or 21.95$ coffee

Step-by-step explanation:

again, divide whichever pair you want to and you still have the same answer whether it is less or more: 450/250 is math would be 9/5 and 21.95/12.35 is 1.77732793522. so if we find the true value of 9/5, which is 1.8, and since it is more that the original price that means the more coffe you get, the cheaper it gets (basically all of life is like this), so the 450 g coffee is worth alot less than and is bigger than the 250 g coffee

The sum of two numbers in 106 and the greater exceeds the lesser in 8. Find the numbers.

Answers

Step-by-step explanation:

51 and 51

maybe

hope it help u

Kern Shipping Inc. has a requirement that all packages must be such that the combined length plus the girth (the perimeter of the cross section) cannot exceed 99 inches. Your goal is to find the package of maximum volume that can be sent by Kern Shipping. Assume that the base is a square.

a. Write the restriction and objective formulas in terms of x and y. Clearly label each.
b. Use the two formulas from part (a) to write volume as a function of x, V(x). Show all steps.

Answers

Answer:

Step-by-step explanation:

From the given information:

a)

Assuming the shape of the base is square,

suppose the base of each side = x

Then the perimeter of the base of the square = 4x

Suppose the length of the package from the base = y; &

the height is also = x

Now, the restriction formula can be computed as:

y + 4x ≤ 99

The objective function:

i.e maximize volume V = l × b × h

V = (y)*(x)*(x)

V = x²y

b) To write the volume as a function of x, V(x) by equating the derived formulas in (a):

y + 4x ≤ 99   --- (1)

V = x²y          --- (2)

From equation (1),

y ≤ 99 - 4x

replace the value of y into (2)

V ≤ x² (99-4x)

V ≤ 99x² - 4x³

Maximum value V = 99x² - 4x³

At maxima or minima, the differential of [tex]\dfrac{d }{dx}(V)=0[/tex]

[tex]\dfrac{d}{dx}(99x^2-4x^3) =0[/tex]

⇒ 198x - 12x² = 0

[tex]12x \Big({\dfrac{33}{2}-x}}\Big)=0[/tex]

By solving for x:

x = 0 or x = [tex]\dfrac{33}{2}[/tex]

Again:

V = 99x² - 4x³

[tex]\dfrac{dV}{dx}= 198x -12x^2 \\ \\ \dfrac{d^2V}{dx^2}=198 -24x[/tex]

At x = [tex]\dfrac{33}{2}[/tex]

[tex]\dfrac{d^2V}{dx^2}\Big|_{x= \frac{33}{2}}=198 -24(\dfrac{33}{2})[/tex]

[tex]\implies 198 - 12 \times 33[/tex]

= -198

Thus, at maximum value;

[tex]\dfrac{d^2V}{dx^2}\le 0[/tex]

Recall y = 99 - 4x

when at maximum x = [tex]\dfrac{33}{2}[/tex]

[tex]y = 99 - 4(\dfrac{33}{2})[/tex]

y = 33

Finally; the volume V = x² y is;

[tex]V = (\dfrac{33}{2})^2 \times 33[/tex]

[tex]V =272.25 \times 33[/tex]

V = 8984.25 inches³

Help pleaseeeee will give brainliest

Answers

Answer:

q.12

angle ACB=180-123

therefore ACB=57

again 5x-15+7x+6+57=180

or,12x+48=180

or,x=132/12

or x=11

A random sample of 30 patties that were inspected over the course of the last week revealed that the average weight was 95.0 grams. The standard deviation was 0.25 grams. What percentage of the deliveries is likely to be outside the specification limits (outside the interval of [94.5, 95.5])

Answers

Answer:

4.56% of the deliveries are likely to be outside the specification limits.

Step-by-step explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

The average weight was 95.0 grams. The standard deviation was 0.25 grams.

This means that [tex]\mu = 95, \sigma = 0.25[/tex]

What percentage of the deliveries is likely to be outside the specification limits (outside the interval of [94.5, 95.5])?

Less than 94.5, or more than 95.5. Since the normal distribution is symmetric, these probabilities are the same, so we can find one of them and multiply by two.

The probability that it is less than 94.5 is the p-value of Z when X = 94.5. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{94.5 - 95}{0.25}[/tex]

[tex]Z = -2[/tex]

[tex]Z = -2[/tex] has a p-value of 0.0228

2*0.0228 = 0.0456

0.0456*100% = 4.56%

4.56% of the deliveries are likely to be outside the specification limits.

Easy question please help

Answers

Answer:

[tex]y = 3x - 2[/tex]

Step-by-step explanation:

Required

The equation of the above linear function

From the table, we have:

[tex](x_1,y_1) = (1,1)[/tex]

[tex](x_2,y_2) = (2,4)[/tex]

Calculate slope (m)

[tex]m = \frac{y_2 -y_1}{x_2 -x_1}[/tex]

[tex]m = \frac{4 -1}{2 -1}[/tex]

[tex]m = \frac{3}{1}[/tex]

[tex]m =3[/tex]

The equation is:

[tex]y = m(x - x_1) + y_1[/tex]

So, we have:

[tex]y = 3(x - 1) + 1[/tex]

[tex]y = 3x - 3 + 1[/tex]

[tex]y = 3x - 2[/tex]


help please quick please

Answers

Answer:

the answer is 3.5

Step-by-step explanation:


[tex] {x}^{2} + \sqrt{x} + \sqrt[5]{x} [/tex]
what is f'(3) of this equation?​

Answers

Answer:

[tex]3 + \frac{1}{2\sqrt{3} } + \frac{1}{5\sqrt[5]{81} }[/tex]

Step-by-step explanation:

Just to make it easier to see, [tex]\sqrt{x} = x^{\frac{1}{2} }[/tex] and [tex]\sqrt[5]{x} = x^{\frac{1}{5} }[/tex]  This way we could more easily use the power rule of derivatives.

So if f(x) = [tex]x^{2} +x^{\frac{1}{2} } +x^{\frac{1}{5} }[/tex] then f'(x) will be as follows.

f'(x) = [tex]x^{1} +\frac{1}{2} x^{-\frac{1}{2} } +\frac{1}{5} x^{-\frac{4}{5} } = x +\frac{1}{2x^{\frac{1}{2} }} +\frac{1}{ 5x^{\frac{4}{5} }} = x +\frac{1}{2\sqrt{x}} +\frac{1}{ 5\sqrt[5]{x^4} }[/tex]

to find f'(3) just plug 3 into f'(x) so [tex]3 + \frac{1}{2\sqrt{3} } + \frac{1}{5\sqrt[5]{81} }[/tex]

Find the distance between a point (–7, –19) and a horizontal line at y = 3.

Answers

Try to use this rule and use those 2 given points——-

(-7,-19) (0,3)

Mention 3 places
where you can get
pre-approved for a
car loan

Answers

Answer:

Auto Credit Express, Carvana, Capital one auto loan

On a coordinate plane, triangle B C D has points (negative 4, 1), (negative 2, 1), (negative 4, 3). Triangle B prime C prime D prime has points (negative 1, negative 4), (negative 1, negative 2), (negative 3, negative 4). Triangle BCD is rotated counterclockwise to form triangle B’C’D’. What is the angle of rotation? 45° 90° 180° 360°

Answers

9514 1404 393

Answer:

  90° CCW

Step-by-step explanation:

The transformation from B to B' is ...

  B(-4, 1) ⇒ B'(-1, -4)

  (x, y) ⇒ (-y, x) . . . . . matches the transformation for 90° CCW

Answer:

90 degrees

Step-by-step explanation:

Solve for z
3z-5+2z=25-5z

Answers

Answer:

z=3

Step-by-step explanation:

1. collect like terms

5z-5=25-5z

2. Move the variable to the left hand side and change its sign

5z-5+5z=25

3. Collect like terms

10z=25+5

4. Divide both sides of the equation by 10

z=3

The solution to the equation is z = 3.

To solve for z in the equation 3z - 5 + 2z = 25 - 5z, we can simplify and combine like terms on both sides:

3z + 2z + 5z = 25 + 5

Combining the terms on the left side gives:

10z = 30

Next, we isolate the variable z by dividing both sides of the equation by 10:

(10z)/10 = 30/10

This simplifies to:

z = 3

Therefore, the solution to the equation is z = 3.

To know more about equation:

https://brainly.com/question/10724260


#SPJ6

Which inequality is true?
А. Зп > 9
B. 7 + 8< 11
C. 27 -1 < 5
D. 2 > 2
SUBMIT
< PREVIOUS

Answers

9514 1404 393

Answer:

  А. Зп > 9

Step-by-step explanation:

The inequality of A may or may not be true. (It is true only if n > 3.) All of the others are definitely false.

In a sample of 400 students, 60% of them prefer eBooks.

A.Find 98% Confidence Interval for the proportion of all students that prefer ebooksb.

b. Find the margin of erro

Answers

Answer:

a) The 98% Confidence Interval for the proportion of all students that prefer ebooks is (0.55, 0.65).

b) The margin of error is of 0.05.

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

z is the z-score that has a p-value of [tex]1 - \frac{\alpha}{2}[/tex].

The margin of error is of:

[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In a sample of 400 students, 60% of them prefer eBooks.

This means that [tex]n = 400, \pi = 0.6[/tex]

98% confidence level

So [tex]\alpha = 0.02[/tex], z is the value of Z that has a p-value of [tex]1 - \frac{0.02}{2} = 0.99[/tex], so [tex]Z = 2.054[/tex].

Margin of error -> Question b:

[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

[tex]M = 2.054\sqrt{0.6*0.4}{400}}[/tex]

[tex]M = 0.05[/tex]

The margin of error is of 0.05.

A.Find 98% Confidence Interval for the proportion of all students that prefer ebooksb.

Sample proportion plus/minus the margin of error.

0.6 - 0.05 = 0.55

0.6 + 0.05 = 0.65

The 98% Confidence Interval for the proportion of all students that prefer ebooks is (0.55, 0.65).

Answer to the question? ​

Answers

Answer:

35

Step-by-step explanation:

AEC and AEB form a straight angle(180°)

180-40=140

AEV and AED are equal

140 divided by 4 = 35

Will choose brainliest! Please help! (This is Khan Academy)

Answers

Answer:

Option B. A = (5/6)^-⅛

Step-by-step explanation:

From the question given above, we obtained:

(5/6)ˣ = A¯⁸ˣ

We can obtain the value of A as follow:

(5/6)ˣ = A¯⁸ˣ

Cancel x from both side

5/6 = A¯⁸

Recall:

M¯ⁿ = 1/Mⁿ

A¯⁸ = 1/A⁸

Thus,

5/6 = 1/A⁸

Cross multiply

5 × A⁸ = 6

Divide both side by 5

A⁸ = 6/5

Take the 8th root of both sides

A = ⁸√(6/5)

Recall

ⁿ√M = M^1/n

Thus,

⁸√(6/5) = (6/5)^⅛

Therefore,

A = (6/5)^⅛

Recall:

(A/B)ⁿ = (B/A)¯ⁿ

(6/5)^⅛ = (5/6)^-⅛

Therefore,

A = (5/6)^-⅛

33. Given the following algebraic expression 5x² + 10 Which statement is true?
a. The coefficient is 5
b. The constant is 2
C. The power is 10
d. The constant is 5

Answers

Answer:

Given the following algebraic expression 5x² + 10 Which statement is true?

a. The coefficient is 5. ( true)

b. The constant is 2

C. The power is 10

d. The constant is 5

the answer is A
hope this helps

The radius of a circle is 10 cm. Find its circumference in terms of \piπ.

Answers

[tex]{ \bf{ \underbrace{Given :}}}[/tex]

Radius of the circle "[tex]r[/tex]" = 10 cm.

[tex]{ \bf{ \underbrace{To\:find:}}}[/tex]

The circumference of the circle.

[tex]{ \bf{ \underbrace{Solution :}}}[/tex]

[tex]\sf\orange{The\:circumference \:of\:the\:circle\:is\:20\:π\:cm.}[/tex]

[tex]\large\mathfrak{{\pmb{\underline{\red{Step-by-step\:explanation}}{\red{:}}}}}[/tex]

We know that,

[tex]\sf\purple{Circumference\:of\:a\:circle \:=\:2πr }[/tex]

[tex] = 2 \: \pi \times 10 \: cm \\ \\ = 20 \: \pi \: cm[/tex]

Therefore, the circumference of the circle is 20 π cm.

[tex]\huge{\textbf{\textsf{{\orange{My}}{\blue{st}}{\pink{iq}}{\purple{ue}}{\red{35}}{\green{♡}}}}}[/tex]

Help me please I NEED to pass this

Answers

OPTION C is the correct answer.

Hope it helps you.

A certain country has 586.08 million acres of forest. Every year, the country loses 7.92 million acres of forest mainly due to deforestation for farming purposes. If this situation continues at this pace, how many years later will the country have only 237.6 million acres of forest left? (Use an equation to solve this problem.)

Answers

Answer:

At this pace the country will have only 237.6 million acres of forest left in 44 years.

Step-by-step explanation:

Given that a certain country has 586.08 million acres of forest, and every year, the country loses 7.92 million acres of forest mainly due to deforestation for farming purposes, to determine, if this situation continues at this pace, how many years later will the country have only 237.6 million acres of forest left, the following calculation must be performed:

Current amount - (amount lost per year x number of years) = 237.6

586.08 - (7.92 x X) = 237.6

586.08 - 7.92X = 237.6

-7.92X = 237.6 - 586.08

-7.92X = -348.48

X = -348.48 / -7.92

X = 44

Therefore, at this pace the country will have only 237.6 million acres of forest left in 44 years.

please help now
Your pump empties the water from a swimming pool in 4 hours. When your friend's pump is used together with your pump, the pool is emptied in 48 minutes. How long (in hours) does it take your friend's pump to empty the pool when working alone?

Answers

Answer:

Time taken for pump B to empty pool = 1 hour.

Step-by-step explanation:

Given:

Time taken for pump A to empty pool = 4 hour

Time taken together = 48 minutes = 48 / 60 = 4/5 hour

Find:

Time taken for pump B to empty pool

Computation:

Assume;

Time taken for pump B to empty pool = a

1/4 + 1/a = 1 / (4/5)

1/4 + 1/a = 5/4

1/a = 5/4 - 1/4

1/a = (5 - 1) / 4

1/a = 1

a = 1

Time taken for pump B to empty pool = 1 hour.

A ramp is in the shape of a triangle

Answers

Nice but what’s the question lol

Answer:

Step-by-step explanation:

Which expression is equivalent to 1/2x + 8

Answers

Answer:

1/2( x+16)

Step-by-step explanation:

1/2x + 8

Factor out 1/2

1/2*x + 1/2 *16

1/2( x+16)

{2x + y = -1}
{3x- 5y + -21}

Answers

Answer:

(-2, 3).

Step-by-step explanation:

2x + y = -1

3x- 5y = -21

Multiply the first equation by 5:

10x + 5y = -5     Adding this to the second equation:

13x = -26

x = -2.

Substituting this into equation 1:

2(-2) + y = -1

y = -1 + 4 = 3.

Checking this result in the second equation:

3(-2) - 5(3)

= -5 - 15 = -21

- Checks OK.

The retail cost of a TV is 50 ​% more than its wholesale cost.​ Therefore, the retail cost is​ ____ times the wholesale cost.

Answers

Answer:

Let the retail cost be x and the wholesale cost be y

Step-by-step explanation:

x = y + 0.50y

x = 1.50y

Therefore the retail cost is 1.50 times the wholesale cost.

An eagle flies towards south luzon at 90 meters for 5 seconds . What is the speed and velocity of the eagle

Answers

Answer:

the speed in 18mph

Step-by-step explanation:

Please help a girl out, math is not my forte

Answers

Answer:

80 ft²

Step-by-step explanation:

You are given the formula

a = (1/2)bh

Just plug in the base and height, then multiply

a = (1/2) * 8 *20

a = (1/2) * 160

a = 80 ft²

Answer:

80 [tex]ft^{2}[/tex]

Step-by-step explanation:

Area = [tex]\frac{1}{2} bh[/tex]

Area = [tex]\frac{1}{2}[/tex] 8 · 20

Area = [tex]\frac{1}{2}[/tex] 160

Area = 80 [tex]ft^{2}[/tex]

Is this the correct answer?

Answers

Answer:

Correct.

Step-by-step explanation:

It looks good to me.

Good job!

Write the word sentence as an inequality.
3.2 less than a number t is at most 7.5

Answers

t-3.2 ≤ 7.5

"at most" means less than or equal to

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