1:Under what condition will the line px+py+r=0 mat be a normal to the circke x²+y²+2gx+2fy+c=0
2:prove that the two circles x²+y²+2ax+c²=0 and x²+y²+2by+c²=0 touch if
[tex]\frac{1}{a²}+\frac{1}{b²}=\frac{1}{c²}[/tex]
Answers
Answer:
#1The normal overlaps with the diameter, so it passes through the center.
Let's find the center of the circle:
x² + y² + 2gx + 2fy + c = 0(x + g)² + (y + f)² = c + g² + f²The center is:
(-g, -f)Since the line passes through (-g, -f) the equation of the line becomes:
p(-g) + p(-f) + r = 0r = p(g + f)This is the required condition
#2
Rewrite equations and find centers and radius of both circles.
Circle 1
x² + y² + 2ax + c² = 0 (x + a)² + y² = a² - c²The center is (-a, 0) and radius is √(a² - c²)Circle 2
x² + y² + 2by + c² = 0 x² + (y + b)² = b² - c²The center is (0, -b) and radius is √(b² - c²)The distance between two centers is same as sum of the radius of them:
d = √(a² + b²)Sum of radiuses:
√(a² - c²) + √(b² - c²)Since they are same we have:
√(a² + b²) = √(a² - c²) + √(b² - c²)Square both sides:
a² + b² = a² - c² + b² - c² + 2√(a² - c²)(b² - c²)2c² = 2√(a² - c²)(b² - c²)Square both sides:
c⁴ = (a² - c²)(b² - c²)c⁴ = a²b² - a²c² - b²c² + c⁴a²c² + b²c² = a²b²Divide both sides by a²b²c²:
1/a² + 1/b² = 1/c²Proved
Answer:
Answer is in the picture
Related Questions
A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $20 per linear foot to install and the farmer is not willing to spend more than $7000, find the dimensions for the plot that would enclose the most area. (Enter the dimensions as a comma separated list.)
Answers
Answer:
The dimensions for the plot that would enclose the most area are as follows:
Width = 116.67 ft
Length = 175 ft.
Step-by-step explanation:
Note: Kindly refer to the attached diagram where:
x = Length of the rectangular area
y = Width of the rectangular area.
The fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. This implies that it is $10 per foot. Otherwise, it will be $20 per foot. Since no fencing is needed along the barn, it implies that there is no fence along the northern length.
Since the farmer is not willing to spend more than $7000, we have:
10y + 20x + 20y = 7000
20x + 30y = 7000
Divide through by 20, we have:
x + 1.5y = 350
x = 350 - 1.5y …………………………. (1)
The fenced area is
A = x * y ………………………………….. (2)
Substituting equation (1) into (2), we have:
A = (350 - 1.5y)y
A = 350y - 1.5y^2 ……………. (3)
To maximize A, equation (3) is differentiated with respect to y and equated it to 0 as follows:
That is,
A'(y) = 350 - 3y = 0
Solving for y, we have:
y = 350/3
y = 116.67 ft
Substituting y = 116.67 for y in equation (1), we have:
x = 350 - (1.5 * 116.67)
x = 175ft
Therefore, the dimensions for the plot that would enclose the most area are as follows:
Width = 116.67 ft
Length = 175 ft.
How many prime numbers less than 20
Answers
Answer:
Prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17 and 19
Step-by-step explanation:
Simply the following ratio 1000:540:780
Answers
Reorder the premises to show that the conclusion follows as a valid consequence from the premises. You may restate them in if-then form or by their contrapositives. (5 pts)
(a) Things that taste good are expensive.
(b) Things that smell good taste good.
(c) Every object to the left of the tree is blue.
(d) All blue objects smell good. :.
Every object to the left of the tree is expensive.
Answers
Step-by-step explanation:
For this question, what we have to do is to rewrite and form some of these statements with their contrapositives
here is the reordering;
1. if Object is at the left of the tree, then it is blue
2. If the object is blue then it smells good.
3. If thus thing smells good then it tastes good.
4. If it tastes good then it is expensive
∴ every object to the left of the tree is expensive
help me pleaseeeeeeeeeeeee
Answers
Answer:
|7000 - w| < 1000
Step-by-step explanation:
6000 < w < 8000
Subtract 7000 from the three "sides."
-1000 < w - 7000 < 1000
|w - 7000| < 1000
|7000 - w| < 1000
the ratio of berries to oranges is 10:1 if there are 25 oranges, how many berries are there?
Answers
Answer:
250 berries
Step-by-step explanation:
berries : oranges
10 1
We have 25 oranges so we need to multiply each side by 25
berries : oranges
10*25 1*25
250 25
There are 250 berries
We can set up the proportion:
(10/1)=(x/25)
Then solve for x
Can you help me with this question? 15 points and brainliest if right!!
Answers
How to take a screenshot on an HP laptop
Press the Windows key and Print Screen at the same time to capture the entire screen. ...
Open an image editing program (Microsoft Paint, GIMP, Photoshop, and PaintShop Pro will all work).
Open a new image and press CTRL + V to paste the screenshot.
The sum of three numbers is 28. The second number is three times the first and the third is 7 less than the second. What are the numbers
Answers
First term: 5
Second: 15
Last : 8
Equation:
x+3x+3x-7=28
Solving:
3x+3x+x=7x
7x-7=28
7x=28+7
7x=35
x=5
Plug 5 into the equations and you’ll get:
First term = 5
Second term = 15
Third term = 8
Distinguish rational numbers from the following.
a. √25 b. √23 c. 1.01 d. 2.323223222... e. 3.786
Answers
Answer:
spongebob at the bottom of the sea
Step-by-step explanation:
In a plain, robust, conversational style, the author known as “Elena Ferrante” has captivated readers worldwide with her chronicle of a complicated friendship between two women.
If the length of EG is 22, find the length of a EQ
Answers
Answer:
A. 11
Step-by-step explanation:
EQ is half of EG
so 22/2 = 11
ANSWER: 11
look at pic 10 pts will mark brainilest zion ihsjzxu
Answers
Answer:
36
Step-by-step explanation:
You first solve 6x=72 which you divide 72/6 and get the answer x=12. You then replace x with 12 and do the order of operations. 12-3 then 9x4.
Answer:
Multiply using the distributive property.
B.
Please mark brainliest!
A piece-wage earner collected the following weights of coffee beans each day over a week:13.3kg,10.8kg,14.4kg and 12,9kg.if she is paid at K5.80 per kilogram, estimate his earning for the week .note do not round off the rate payment
Answers
Answer:
14.4 should be around 14kg
12,9kg should be rounded to 12
ally and bo share 60$ in the ratio 1:4 how much money does bo get
Answers
Answer: $240
Step-by-step explanation:
So 60 represents the 1, so multiply 60 by 4
In a sample of 230 teenagers, 202 would like to have smaller class sizes at their school. Find the sample
proportion, the margin of error, and the interval likely to contain the true population proportion. If
necessary, round your answers to the nearest percent.
Answers
Answer:
The sample proportion is 0.88.
The margin of error is of 0.04.
The interval likely to contain the true population proportion is (0.84,0.92).
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].
Margin of error:
The margin of error is of:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
Sample of 230 teenagers, 202 would like to have smaller class sizes at their school.
This means that [tex]n = 230[/tex], and that the sample proportion is:
[tex]\pi = \frac{202}{230} = 0.88[/tex]
The sample proportion is 0.88.
Standard 95% confidence level
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a p-value of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
Margin of error:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]M = 1.96\sqrt{\frac{0.88*0.12}{230}}[/tex]
[tex]M = 0.04[/tex]
The margin of error is of 0.04.
Confidence interval:
Sample proportion plus/minus margin of error. So
0.88 - 0.04 = 0.84
0.88 + 0.04 = 0.92
The interval likely to contain the true population proportion is (0.84,0.92).
at 90kph, how far do you travel in 6 hours
Answers
Answer:
540 km
Step-by-step explanation:
You find distance by multiplying rate and time. Rate is 90, and time is 6. 90 x 6 = 540 km
40 POINTS! Match each set of vertices with the type of triangle they form.
Answers
Answer:
#1 ((2, 0), (3, 2), (5, 1)) = right isoceles triangle ✔️
#2 ((4, 2), (6, 2), (5, 3.73)) = acute isosceles triangle ❌
so it is not part of graph
#3 ((-5, 2), (-4, 4), (-2, 2)) = Acute scalene triangle ✔️
#4 ((-3, 1), (-3, 4), (-1, 1)) = Right scalene triangle (or right triangle)✔️
#5 ((-4, 2), (-2, 4), (-1, 4)) = Obtuse scalene triangle ✔️
Step-by-step explanation:
the triangles are correct
Answer:
(2,0), B(3,2),C(5,1) is isosceles right angel
(-5,2), B(-4,4), C(-2,2) acute scalene triangle
(-3,1), B(-3,4), C(-1,1) right angel
(-4,2), B9-2,4), C(-1,4) obtuse scalene triangle
Step-by-step explanation:
Find the coordinates of the points that are 20 units away from the origin and have a y-coordinate equal to −12.
Answers
9514 1404 393
Answer:
(-16, -12), (16, -12)
Step-by-step explanation:
The given values (side length and hypotenuse) have the ratio ...
12 : 20 = 3 : 5
This suggests that the triangle formed by the axes and the point 20 units from the origin is a 3-4-5 triangle, and the remaining side is 16 units from the y-axis. That means the points of interest are ...
(-16, -12) and (16, -12)
The pH scale measures how acidic or basic a substance is. Lemon juice is said to have a pH of less than 4 and greater than 1.5. Model the normal range of pH values of lemon juice, using a compound inequality.
1.5 > x > 4
1.5 < x < 4
1.5 ≤ x ≤ 4
1.5 ≥ x ≥ 4
Answers
Answer: 1.5 < x < 4
Step-by-step explanation:
Line segment CD is the altitude drawn to hypotenuse EF in right triangle ECF. If EC = 10 and EF = 24, then, to the nearest tenth, find ED.
Answers
Answer:
4.2 = ED
Step-by-step explanation:
C is the Right Angle.You will do yourself a great favor by drawing your own diagram. The easiest way to do this is to find CF first. Use the Pythagorean Theorem.The next step is to find the area of the triangles using EC * CF which you just found. Equate that to the hypotenuse * altitude(CD) to find CDThen use Pythagoras again to ED.Step One
Find FC. That is the only side of the right triangle that you don't know the value of.
EC^2 + FC^2 = EF^2EC = 20FC = ?EF= 24 This the Hypotenuse.10^2 + FC^2 = 24^2100 + FC^2 = 576 Subtract 100 from both sides.FC^2 = 576 - 100 Combine the right FC^2 = 476sqrt(FC^2) = sqrt(476)FC = 21.817Step Two
Find CD. Do this by finding the area two ways. (Key Step)Area = hypotenuse * altitude / 2Area = EC * CF / 2EC = 10CF = 21.817hypotenuse = 24altitude = y24*y / 2 = 10 * 21.817 /212 y = 218.17/212y = 109.085y = 109.085/12y = 9.0904Step Three
Find EDED^2 = 10^2 - y^2ED^2 = 100 - 9.0904^2ED^2 = 100 - 82.6357ED^2 = 17.36sqrt(ED^2) = sqrt(17.36)ED = 4.1671ED = 4.2The box plots below show the math scores of students of two different classes, which statement is correct
Answers
Answer:
B
Step-by-step explanation:
Given the data:
Class A 55 72 75 89 95
Class B 55 70 75 94 100
CLASS A :
Minimum - 55
Lower quartile - 72
Median - 75
Upper quartile - 89
Maximum - 95
CLASS B :
Minimum - 55
Lower quartile - 70
Median - 75
Upper quartile - 94
Maximum - 100
Evaluating the statements given :
The median score of Class A is greater than the median score of Class B.
Median A = 75 ; Median B = 75 (median are equal), hence, A is not true
The lower quartile of Class A is greater than the lower quartile of Class B.
Lower quartile A = 72 ; Lower quartile B = 70 (A is greater than B) , hence, B is true
The upper quartile of Class A is greater than the upper quartile of Class B.
Upper quartile A = 89 ; Upper quartile B = 94 (A is less than B) , hence, statement is not true
The maximum score of Class A is greater than the maximum score of Class B.
Maximum A = 94 ; maximum B = 100 ; A < B ; Hence, statement is not true
Find the volume of the prism.
Answers
Volume of prism= 3.024 yd^3
Explanation: Volume= L x W x H
Find the area of a regular
polygon with 6 sides that has a
perimeter of 42 inches and an
apothem of 5 inches,
Answers
Answer:
The area of a regular polygon is A = (1/2)ap, where a is the apothem and p is the perimeter of the polygon.
The apothem is 8 inches and the perimeter is 6(7) = 42 since there are 7 sides of 6 inches.
Then the area is
A = (1/2)(8)(42) = 336/2 = 168 in2
I hope this helped!
Step-by-step explanation:
Mark me as brainlist pleasee
Brainliest, 10 Points, & Thanks if answer to these questions!!!!!!!!
Answers
Answer:
D
Step-by-step explanation:
+1 and -1 change the x postion
A factory manufactures motorcycles. One of its employees, working in the quality control department, checks the first 10 and the last 10 motorcycles manufactured in a day. This is what type of sampling?
Answers
Answer:
Convenience sampling
Step-by-step explanation:
When sampling is done based in degree of ease or samples picked from observations that are easily accessible rather than prioritizing randomness or selection which will most be representative of the population, then such sampling is called convenience sampling. In some texts, it is also referred to as grab sampling as researchers choose from close, easy to reach samples. In the scenario above, the quality control takes the easy procedure of checking the first and last 10 motorcycles as this is easier than having to take samples at random tune intervals which will be a better representation of the entire motorcycles.
Can somebody help me with this question
Answers
Answer:
y = -⅓x + 7
Step-by-step explanation:
Given the two points that lie on the line (3, 6) and (-3, 8), we can find the equation without the grid by finding the slope (m) and y-intercept (b).
The equation would be written in the form of y = mx + b (slope-intercept form):
✔️Find m:
Slope (m) = change in y/change in x
Slope using (3, 6) and (-3, 8), would be:
Slope (m) = (8 - 6)/(-3 - 3) = 2/-6
m = -⅓
✔️Find b:
Substitute (x, y) = (3, 6) and m = -⅓ into y = mx + b to find b
6 = -⅓(3) + b
6 = -1 + b
6 + 1 = b
b = 7
✅Write the Equation:
Substitute m = -⅓ and b = 7 into y = mx + b
Thus,
y = -⅓x + 7
if gh¢400.00 is deposited at a bank at 10 per annum compounded quarterly, how long will it take the deposit to double it value.
Answers
Answer:
~ 7 years
Step-by-step explanation:
2 = (1 + .1/4)^(4t)
2 = (1.025)^(4t)
Take the log of both sides
log2 = 4t * log1.025
divide both sides by log1.025
log2/log1.025 = 4t
28.07103452593863 = 4t
Divide both sides by 4
7.017758631484657 = t
~ 7 years
Jeff says the figure below is a rhombus. Is he correct? *
Answers
Answer:
yes it is
Step-by-step explanation:
it is quadrilateral
Answer:
yes
Step-by-step explanation:
just look up rhombus and then look at the pictures
If f(x) = x/2 -3 and g(x) = 3x² + x - 6, find (f + g)(x). O A. 3x O B. 3x² + 3x-9 3x O C. 3.x2 + x - 3 O D. 4x2-9
Answers
Answer:
4x2-9
Step-by-step explanation:
it would be that because the number is in the c acis
Find the value of x to the nearest degree.
16
7
Not drawn to scale
x°
Answers
Answer:
66°
Step-by-step explanation:
Given the right angled triangle :
Opposite side = 16
Adjacent side = 7
Using trigonometric relation :
Tan θ = opposite / Adjacent
Tan x = (16/7)
x = tan^-1(16/7)
x = 66.37°
x = 66°
1.05^x=2
Solve for x
Answers
1.05^x = 2
Apply export rule ( logarithm s)
Xln1.05 = ln2
Divide both sides by ln1.05
X = ln2/ln1.05
X = 14.206
Round the answer as needed.