B) there is a line with positive slope that is tangent to both circles. find the equation of this line, and (c) determine the points at which this tangent line touches each circle

Answers

Answer 1

Given that there is a line with positive slope that is tangent to both circles [tex]x^2 + y^2 = 1[/tex] and [tex](x-3)^2 + y^2 = 4[/tex].

Let the point of tangency in the first circle be (a, b) and the point of tangency in the second circle be (c, d), then the slope of the tangent to the first circle is given by

[tex]2x+\left.2y \frac{dy}{dx}\right|_{(a, b)} =0 \\ \\ \Rightarrow \left.\frac{dy}{dx}\right|_{(a, b)} =- \frac{x}{y} =- \frac{a}{b} [/tex]

and the slope of the second circle is given by

[tex]2(x-3)+\left.2y \frac{dy}{dx}\right|_{(c, d)} =0 \\ \\ \Rightarrow \left.\frac{dy}{dx}\right|_{(a, b)} =- \frac{x-3}{y} =- \frac{c-3}{d}[/tex]

From the first circle equation, we have

[tex]x^2 = 1 - y^2 \\ \\ \Rightarrow a^2 = 1 - b^2[/tex]

and from the second equation, we have

[tex](x-3)^2 = 4 - y^2 \\ \\ (c-3)^2 = 4 - d^2[/tex]

Since the two slopes above are for the same line, then

[tex] \frac{a}{b}= \frac{c-3}{d} \\ \\ \Rightarrow ad=b(c-3) \\ \\ \Rightarrow a^2d^2=b^2(c-3)^2 \\ \\ \Rightarrow(1-b^2)d^2=b^2(4-d^2) \\ \\ \Rightarrow d^2-b^2d^2=4b^2-b^2d^2 \\ \\ \Rightarrow d^2=4b^2 \\ \\ \Rightarrow d=\pm2b[/tex]

Thus, we have

[tex] \frac{a}{b}= \frac{c-3}{2b} \\ \\ \Rightarrow a= \frac{c-3}{2} [/tex]

Related Questions

What relationship do the ratios of Sin X and Cos y share ?

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The sine of the measure of an angle in a right triangle, is the ratio of the opposite side of that angle, to the hypotenuse.

The cosine of the measure of an angle in a right triangle, is the ratio of the adjacent side of that angle, to the hypotenuse.

According to these, we have:

[tex]\displaystyle{ \sin(x^{\circ})= \frac{6}{10} [/tex]

and

[tex]\displaystyle{ \sin(y^{\circ})= \frac{6}{10} [/tex].

Thus, the ratios of both are identical [tex]( \frac{6}{10} \ and \ \frac{6}{10} )[/tex]

The ratios of sin (x) and cos (y) are both identical [tex](\frac{6}{10}\; and\; \frac{6}{10})[/tex].

RIGHT TRIANGLE

A triangle is classified as a right triangle when it presents one of your angles equal to 90º. A math tool applied for finding angles or sides in a right triangle is trigonometric ratios.

The main trigonometric ratios are:

[tex]sin(\alpha )=\frac{opposite\;side}{hypotenuse} \\ \\ cos(\alpha )=\frac{adjacent\;side}{hypotenuse}\\ \\ tan(\alpha )=\frac{sin (\alpha )}{cos(\alpha } =\frac{opposite\;side}{adjacent\;side}[/tex]

The question asks the sin x and cos y, for solving this you should apply the trigonometric ratios. Therefore,

[tex]sin (x)=\frac{opposite\; side}{hypotenuse} =\frac{6}{10}[/tex]

and

[tex]cos (y)=\frac{adjacent\; side}{hypotenuse} =\frac{6}{10}[/tex]

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"example 1: the radius of a circle is 3 inches. what is the area?"

Answers

= 28.26 in thats your answer

Algebra help me please ILL MARK BRAINIEST

Answers

The solution is optio D.

0     4
- 6     1
3    -4

Rember row times columns.

=>
a11 = 4*0 + 0*(-3) = 0

a12 = 4*1 + 0*1 = 4 + 0 = 4

a21 = -1*0 + 2*(-3) = 0 - 6 = -6

a22 = -1*1 + 2*1 = -1 + 2 = 1

a31 = -3*0 + (-1)*(-3) = 0 + 3 = 3

a32 = -3*1 + (-1)*1 = -3 - 1 = -4

During strenuous exercise, an athlete can burn 12 calories per minute on a treadmill and 8.5 calories per minute on an elliptical machine. If an athlete uses both machines and burns 325 calories In a 30- minute workout, how many minutes does the athlete spend on each machine?

Answers

Hope it helps 10 mins on the elliptical and 12 on the treadmil

The athlete spends 20 minutes on the treadmill and 10 minutes on the elliptical machine. This is determined by setting up and solving a system of linear equations based on the calorie burn rate of each machine and the total calories burned during the workout.

To solve the problem of how many minutes an athlete spends on each machine during their workout, we must set up a system of linear equations. The athlete burns 12 calories per minute on a treadmill and 8.5 calories per minute on an elliptical machine. In a 30-minute workout, the athlete burns a total of 325 calories. Let's denote the time spent on the treadmill as t minutes and the time spent on the elliptical machine as e minutes.

The total time spent on both machines is 30 minutes: t + e = 30The total calories burned is 325: 12t + 8.5e = 325

We now have two equations and two unknowns, which can be solved simultaneously.

Subtract the second equation from 12 times the first equation to eliminate eSolve the resulting equation for t to find the number of minutes on the treadmillSubstitute the value of t in the first equation to find the number of minutes on the elliptical machine

Using these steps, we find that:

12t + 12e = 360 (by multiplying the first equation by 12)12t + 8.5e = 325 (second equation as is)Subtracting these equations, we get 3.5e = 35Solving for e, we find that e = 10 (the athlete spends 10 minutes on the elliptical machine)Plugging e = 10 into the first equation, we find that t = 20 (the athlete spends 20 minutes on the treadmill)

Therefore, the athlete spends 20 minutes on the treadmill and 10 minutes on the elliptical machine during their 30-minute workout.

Complete each statement regarding an angle θ and its reference angle theta hat using radian measure. (Assume 0 ≤ θ < 2π. Give your answers in terms of θ.) (a) If θ is in QI, then theta hat = . (b) If θ is in QII, then theta hat = . (c) If θ is in QIII, then theta hat = . (d) If θ is in QIV, then theta hat = .

Answers

The reference angle is simply the acute angle between the terminal side and the x-axis

The terminal angles in the four quadrants are

[tex]\bar \theta = \theta[/tex].[tex]\bar \theta = \pi - \theta[/tex].[tex]\bar \theta = \theta - \pi[/tex].[tex]\bar \theta = 2\pi - \theta[/tex]

The given parameters are:

[tex]\theta \to[/tex] reference angle

[tex]\bar \theta \to[/tex] terminal angle

(a) If the reference angle is in the first quadrant

In the first quadrant, the x-axis is at [tex]0\ rad[/tex]

So, the terminal angle is:

[tex]\bar \theta = \theta - 0[/tex]

[tex]\bar \theta = \theta[/tex]

(b) If the reference angle is in the second quadrant

In the second quadrant, the x-axis is at [tex]\pi \ rad[/tex],

While the reference angle is between [tex]\frac{\pi}{2}[/tex] and [tex]\pi \ rad[/tex]

It means that, we have to subtract the reference angle from the x-axis

So, the terminal angle is:

[tex]\bar \theta = \pi - \theta[/tex]

Here, the x-axis is still at [tex]\pi \ rad[/tex],

But the reference angle is between [tex]\pi \ rad[/tex] and [tex]\frac{3}{2}\pi[/tex]

It means that, we have to subtract the x-axis from the reference angle.

So, the terminal angle is:

[tex]\bar \theta = \theta - \pi[/tex]

(d) If the reference angle is in the fourth quadrant

Here, the x-axis is at [tex]2\pi \ rad[/tex]

And the reference angle is between [tex]\frac{3}{2}\pi[/tex] and [tex]2\pi \ rad[/tex]

It means that, we have to subtract the reference angle from the x-axis.

So, the terminal angle is:

[tex]\bar \theta = 2\pi - \theta[/tex]

See attachment for illustration of terminal angles

Answers

at a discounted price

1. at a discount

2. add all the items together at their sales cost then add salves tax

3. $26.75

4. $66.23

Michael sold his house for $548,900, which was 160% of the amount he paid for it. Calculate the amount he paid for the property.

Answers

That would be 548,900 / 1.6 = $343,062.50

A scuba diver swimming at the Deppe shown, and then swim 0.5 feet toward the surface every three seconds. What is the location of the scuba diver, relative to the surface, after 15 seconds?

Answers

For every 3 seconds, the scuba diver dives 0.5 feet
after 15 seconds = your answer

3 x 5 = 15
0.5 x 5 = your answer

your answer = 2.5

2.5 is your answer

hope this helps

anyone help me please!
A stereo listed for $350 is on sale for 20% off the list price. The sales tax is 6%. What total price does a customer pay for the stereo if she/he buys it on sale?
1 $324.00
2 $259.00
3 $262.20
4 $296.80
5 $301.00

Answers

The answer would be 296.80.Hope this helps!

20% subtracted from $350 leaves you with $280

then you need add 6% onto the price

Im not sure if youre supposed to calculate 6% of the original price or the price with 20% off

tax of normal price + discounted price= $301
tax of discounted price + discounted price= $296.60

Your answer should be E since $301 is an answer choice

45+5/6x=50
need to find x and math for x

Answers

45 + 5/6x = 50

5/6x = 50 - 45

5/6x = 5

5x = 5 * 6

5x = 30

x = 6

hope that helps, God bless!

The solution of expression is, x = 6

We have to give that,

An expression to simplify as,

⇒ 45 + 5/6x = 50

Now, Simplify the expression by combining like terms,

⇒ 45 + 5/6x = 50

Subtract 45 on both sides,

⇒ 45 + 5/6x - 45 = 50 - 45

⇒ 5/6x = 5

Multiply by 6 on both sides,

⇒ 5x = 5 × 6

⇒ 5x = 30

Divide by 5 on both sides,

⇒ x = 30/5

⇒ x = 6

So, The solution is, x = 6

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A car is driving at the speed of 58 km/h.How much time will it take to drive for distance 32 miles?

Answers

53 Minutes and 16 seconds

What is 129,543 rounded to the nearest ten thousand

Answers

129,543 rounded to the nearest ten thousand is 130,000

hope this helps

129,543 rounded to the nearest ten thousand is 130,000. Hope this helps! Can u plz mark me as brainliest? I really need it !

Caroline bought 20 shares of stock at 10 ½, and after 10 months the value of the stocks was 11 ¼. If Caroline were to sell all her shares of this stock, how much profit would she make? A. $210 B. $10 C. $15 D. $225

Answers

20 shares at 10 1/2 (or 10.5) = 20 * 10.5 = 210
20 shares at 11 1/4 (or 11.25) = 20 * 11.25 = 225

so the profit made would be : 225 - 210 = $ 15 <==

How much greater is the area of a square with a side length of 9 inches than the area of a circle with a radius of 3 inches?

Answers

The area of the square is 81 sq in, and the area of the circule is (3.14)(3 in)^2, or approx 3.14(9) sq in, or approx 28.27 sq in.

Subtract: 81 sq in - 28.27 sq in = approx. 52.73 sq in.

The difference is 52.73 sq in; the square is larger, the circle smaller.

family drinks 1 3/4 gallons of milk every 4 1/2 days what is the unit rate of milk per day

Answers

[tex]\bf \cfrac{milk}{day}\qquad \cfrac{1\frac{3}{4}}{4\frac{1}{2}}\implies \cfrac{\frac{1\cdot 4+3}{4}}{\frac{4\cdot 2+1}2}\implies \cfrac{\quad \frac{7}{4}\quad }{\frac{9}{2}}\implies \cfrac{7}{\underline{4}}\cdot \cfrac{\underline{2}}{9}\implies \cfrac{7}{2}\cdot \cfrac{1}{9}\implies \cfrac{7}{18}[/tex]

which gives us a quotient of about 0.39 gallons of milk for one day.

a large bag of cashews weighs 8 1/3 pounds one serving is 1/3 pound how many servings are in the bag

Answers

A large bag of cashews weighs is 8(1/3) pounds = which is 25/3 in improper fraction form
one serving bag is 1/3
now, how many servings are in the bag?
=> 25/3 ÷ 1/3
=> 25/3 X 3/1 (changing division into multiplication form)
=> 25, bag are in serving .

What is the product in simplest form?

−89⋅56

Answers

The product would be -4984.

The answer in siplest form is -

Reina and Sam are multiplying (84)5 (73)9. Reina's Work Sam's Work (84)5 (73)9 = 84 + 573 + 9 = 89712 (84)5 (73)9 = 84⋅573⋅9 = 820727 Is either of them correct? Explain your reasoning.

Answers

None of this above 84⋅573⋅9 84*5*73*9 ( )= multiplication all of them should multiply

What percent is 6 out of 30? 2% 6% 20% 30%

Answers

6 is 20 percent of 30

The answer is 20%. Hope I helped

Find the ratio and simplify. In this triangle, find the ratio of the shortest side to the longest. A triangle has three sides with lengths 15.2, 5.6, 14.1

Answers

well, the shortest will be 14.1 clearly, and the longest of all three is 15.2.

let's check the ratio from shortest to longest.

[tex]\bf 14.1\implies \cfrac{141}{10}\qquad \qquad\qquad 15.2\implies \cfrac{152}{10} \\\\\\ \cfrac{shortest}{longest}\qquad 14.1:15.2\qquad \cfrac{14.1}{15.2}\implies \cfrac{\quad \frac{141}{10}\quad }{\frac{152}{10}}\implies \cfrac{141}{10}\cdot \cfrac{10}{152} \\\\\\ \cfrac{141}{152}[/tex]

What value of x makes the equation below true?

6x - 9 = 39

A. 13

B. 5

C. 8

D. 24

Answers

The answer is C. eight

6x - 9 = 39
6x = 48
x = 8

hope this helps , give brainliest

[tex] \frac{dx}{dt} = \frac{t^{2}+3tx+ x^{2} }{ t^{2} } [/tex]

Answers

[tex]\dfrac{\mathrm dx}{\mathrm dt}=\dfrac{t^2+3tx+x^2}{t^2}[/tex]
[tex]\dfrac{\mathrm dx}{\mathrm dt}=1+\dfrac{3x}t+\dfrac{x^2}{t^2}[/tex]

Let [tex]x(t)=ty(t)[/tex], so that [tex]\dfrac{\mathrm dx}{\mathrm dt}=t\dfrac{\mathrm dy}{\mathrm dt}+y[/tex].

[tex]t\dfrac{\mathrm dy}{\mathrm dt}+y=1+3y+y^2[/tex]
[tex]t\dfrac{\mathrm dy}{\mathrm dt}=1+2y+y^2=(1+y)^2[/tex]
[tex]\dfrac{\mathrm dy}{(1+y)^2}=\dfrac{\mathrm dt}t[/tex]
[tex]\implies-\dfrac1{1+y}=\ln|t|+C[/tex]
[tex]\implies y+1=-\dfrac1{\ln|t|+C}[/tex]
[tex]\implies yt=x=-t-\dfrac t{\ln|t|+C}[/tex]

F=J+Jrt solve following formula for t

Answers

F=j+jrt
F-j=jrt
t=(F-j)/jr

Final answer:

To solve the formula F=J+Jrt for 't', first, subtract 'J' from both sides to get F-J=Jrt. Then, divide both sides by 'Jr', for the final equation for 't' = (F-J)/Jr.

Explanation:

This physics-related mathematics question is asking you to solve the given formula F=J+Jrt for 't'. To isolate 't', we first want to eliminate 'J' from the right side. We do this by subtracting 'J' from both sides of the equation. This gives us F-J=Jrt.

Then, we divide both sides by 'Jr'. When we do this, we are left with t=(F-J)/Jr as the rearranged formula with 't' as the subject. This results in 't' being isolated in the equation, which answers the student's question.

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A section of a roller coaster track forms a parallelogram ABCD. If m∠ABC = 72°, what is m∠DAB? A) 72° B) 108° C) 144° D) 216°

Answers

check the picture below.

Answer: Option 'B' is correct.

Step-by-step explanation:

Since we have given that

A section of a roller coaster track forms a parallelogram ABCD.

If m∠ABC = 72°

We need to find the m∠DAB,

Since ABCD is a parallelogram .

So, "Sum of interior angles on the same side is supplementary":

[tex]m\angle ABC+m\angle DAB=180\textdegree\\\\72\textdegree+m\angle DAB=180\textdegree\\\\m\angle DAB=180\textdegree-72\textdegree\\\\m\angle DAB=108\textdegree[/tex]

Hence, Option 'B' is correct.

a woman walks in and steals 100 from a register. 5 minutes later uses the same 100 to buy merchandise from that same store and buys 70 worth of merchandise with that same 100. he gives her 30 in change. how much did the store lose

Answers

she took 100, the store has - 100
she went back and bough
70 bucks in stuff, the store has - 70

yes, she took the $100 bill and gave it, got back $30, that means she only kept $30 from the original bill.

the store however, lost the original $100 bill plus the $70 in merchandise, so they have -$170.

A driveway is 40 yards long. About how many meters long is it? Round to the nearest hundredth

Answers

The driveway is 36.576 meters long.

How many terms are in the arithmetic sequence 7, 0, −7, . . . , −175?

Answers

so hmmm 7, 0, -7.... what the dickens is going on? hmmm is really dropping each time by 7, so, 7-7, 0, and 0-7, -7 and so on.

so, the "common difference" is then -7, and our first term is 7, now, who's -175? let's check.

[tex]\bf a_n=a_1+(n-1)d\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ d=-7\\ a_1=7\\ a_n=-175 \end{cases} \\\\\\ -175=7+(n-1)(-7)\implies -175=7+7-7n \\\\\\ 7n=14+175\implies 7n=189\implies n=\cfrac{189}{7}\implies n=\stackrel{terms}{27}[/tex]

What is the simplified expression for the expression below? 1/2(8x+4)+1/3(9-3x)

Answers

1/2(8x + 4) + 1/3(9 - 3x)

Simplify.

4x + 2 + 3 - x

Combine like terms.

(4x - x) + (2 + 3)

Simplify.

3x + 5

~Hope I helped!~

Answer:

3x + 5

Step-by-step explanation:

A pro shop in a bowling center decreases the price of a urethane bowling ball by 22% to a sale price of $88.14. What is the former price

Answers

88.14/.78=$113 is you Answer

1x - .22x = 88.14
.78x = 88.14

divide by .78 x = 113

Jenny drove 1,255 miles last week. She drove 187 miles on Monday. On Tuesday, she drove 53 more miles than on Monday, on Wednesday she dove 26 more miles than on Tuesday. How many more miles did Jenny drove on Monday and tuesday together than on Wednesday and Thursday together?

Answers

The answer should be 401

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