Answers
Step-by-step explanation:
Given:
The three points given below are collinear,
[tex](x_1,y_1) = (a,0)\\(x_2,y_2) = (0,b)\\(x_3,y_3) = (1,1)[/tex]
To prove:
[tex] \frac{1}{a} + \frac{1}{b} = 1[/tex]
Proof:
Collinear: given set of points lies on a straight line.
Now we are told that the given set of points are collinear hence they will not be able to form a triangle or the area enclosed by all the points will be equal to zero.
The formula of area of traingle in determinant form is
[tex]Area = \frac{1}{2} \begin{vmatrix} x_{1}& y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{vmatrix}[/tex]
that can also be written as
[tex] \sf \small Area = \frac{1}{2} [x_1 (y_2 - y_3) + x_3 (y_1 - y_2) + x_2 (y_3 - y_1)][/tex]
Since the points are collinear L.H.S will be zero,
[tex]\sf \small 0 = \frac{1}{2} [x_1 (y_2 - y_3) + x_3 (y_1 - y_2) + x_2 (y_3 - y_1)][/tex]
Substituting all the given cordinates in above equation.
[tex]\sf \small 0 = \frac{1}{2} [a (b - 1) + 1 (0 - b) + 0 (1 - 0)] [/tex]
[tex]\sf \small 0 \times 2 = ab - a - b \\ \sf \small 0 = ab - a - b \\ \sf \small a + b = ab \\ \sf \small \: dividing \: both \: sides \: by \: ab \\ \sf \frac{a}{ab} + \frac{b}{ab} = \frac{ab}{ab} \\ \sf \frac{\cancel a}{\cancel ab} + \frac{\cancel b}{a \cancel b} = \cancel \frac{ab}{ab} \\ \sf \frac{1}{a} + \frac{1}{b} = \frac{1}{1} [/tex]
[tex] \sf Hence \: proved,[/tex]
[tex] \Large {\frac{1}{a} + \frac{1}{b} = 1}[/tex]
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Use area of triangle formed by using those points is equal to zero
[tex] \tt Area\: of \:triangle =\frac{1}{2}[a(b-1)+0+1(0-b)][/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: [/tex]
[tex] \tt0=ab-a-b[/tex]
[tex]\: \: \: \: \: \: \: \: \: \: \: [/tex]
[tex] \tt a+b=ab[/tex]
Divide both sides by ab
[tex] \tt \frac{a}{ab} + \frac{b}{ab} = 1[/tex]
[tex]\: \: \: \: \: \: \: \: \: \: \: [/tex]
[tex] \tt\frac{1}{b} + \frac{1}{a} = 1[/tex]
Hence proved~
Related Questions
Solve the problem.
A recipe calls for 6 ounces of unsweetened chocolate for 20 brownies. How much of unsweetened chocolate is needed to make 30 brownies?
Answers
3
6
2
•
3
12
O y=2x+:
O y = 3x
O y =3x+3
O y=x+3
Which equation represents the relationship shown in the table? •
Answers
Answer:
y = 3x + 3
Step-by-step explanation:
when you check, only that equation delivers the correct x and y pairs.
x = 0, => y = 3×0 + 3 = 3
x = 1, y = 3×1 + 3 = 3+3 = 6
x = 2, y = 3×2 + 3 = 6 + 3 = 9
x = 3, y = 3×3 + 3 = 9 + 3 = 12
it all fits.
The length of AB is 9 centimeters. A dilation with a scale factor of 2 is applied to AB. What is the length of the image AB after the dilation is applied?
Answers
Answer:
18cm
Step-by-step explanation:
Given:
- The length of AB is 9 centimeters
- A dilation with a scale factor of 2
The length of the image AB after the dilation is applied:
9 x 2 = 18
dilation is used to make an image or in this case AB, with a scale factor of 2 means that its doubling in size. If the scale factor was 1/2 that it would be decreasing in size since it will be half of the original size
Write an equation of the line that passes through the point (6,-9) with the given slope m= 3
Answers
Answer:
y=3x-27
Step-by-step explanation:
[tex]\rule{300}{1}\\\dashrightarrow\large\blue\textsf{\textbf{\underline{Given question:-}}}[/tex]
Write an equation of the line passing through (6, -9) and has a slope of -3.
[tex]\dashrightarrow\large\blue\textsf{\textbf{\underline{Answer and how to solve:-}}}[/tex]
First, we need to write the equation in point-slope form:-
[tex]\hookrightarrow\sf{y-y_1=m(x-x_1)}[/tex]
Replace y₁ with -9, m with 3, and x₁ with 6:-[tex]\hookrightarrow\sf{y-(-9)=3(x-6)}[/tex]
[tex]\bigstar[/tex] On simplification,
[tex]\hookrightarrow\sf{y+9=3(x-6)}[/tex]
[tex]\bigstar[/tex] On further simplification,
[tex]\hookrightarrow\sf{y+9=3x-18}[/tex]
[tex]\star[/tex] Subtract 9 from both sides, which results in:-
[tex]\hookrightarrow\sf{y=3x-27}[/tex]
So we conclude that Option B is correct.
Good luck with your studies.
[tex]\rule{300}{1}[/tex]
The population of a city decreases by 4.4% per year. If this year's population is
135,000, what will next year's population be, to the nearest individual?
Answers
Answer:
129060
Step-by-step explanation:
135 000 - .044(135000) = 129 060
A(5, 10) and B(13,-2) are two points on the line AB.
The perpendicular bisector of the line AB has gradient
Find the equation of the perpendicular bisector of AB
Answers
Answer:
hope you understand......
Give the value of x .
A. 60 degrees
B. 180 degrees
C. 120 degrees
D. 45 degrees
Answers
Answer:
x=120°
Step-by-step explanation:
sum of all angles in a triangle is equal to 180
let the unknown angle be
[tex] \alpha [/tex]
[tex]60 + 60 + \alpha = 180 \\ \alpha = 180 - 60 - 60 \\ \alpha = 60 \\ [/tex]
[tex] \alpha + x = 180 \\ x = 180 - \alpha \\ x = 180 - 60 \\ x = 120[/tex]
Help me please I need this done quick
Answers
1058.4 in^2
Step-by-step explanation:
Find the surface areas of the rectangular prism and the triangular prisms separately.
Triangular: S = (1/2)lP+B, where l is slant height, P perimeter, and B base area.
14(4)= 56 perimeter of base
13 slant height
B = 14x14 = 196
put together:
S = (1/2)(13 x 56) + 196
S = 560 in^2
Now the rectangular prism
S = 2lw + 2lh + 2wh, where l is length, h height, w width. (delete the first 2lw since they share one side/they're combined shapes.
S = 2(14x8.9) + 2(14x8.9)
S = 498.4 in^2
Add them together: 498.4 + 560 = 1058.4 in^2
BOOM Creations has offered to sell its balloon
bouquets for a fixed down payment of $65 and an
additional charge of $2.75 per bouquet.
Answers
Answer:
y intercept: 65
rate of change: $2.75
Step-by-step explanation:
Really quite simple lad. Your equation is y=2.75x+65. 65 is your y-intercept, and you have a rate of change of 2.75 per bouquet.
helppppppppppppppppppppppppppppppppppppppppppppp
Answers
So you divide the price by the unit (kg)
And find the better buy by looking at which price is cheapest.
Your answer is the 7.48 kg bag
Hope this helps and good luck!!
Consider a game in which you draw playing cards from a standard 52-card deck, one at a time. If a player draws a face card (a jack, a queen, or a king), the player is awarded 10 points. Any other card drawn costs the player 3 points. What is the expected value of drawing a card in this game? A. 0 points B. 3 points C. 6.5 points D. 10 points
Answers
Answer: B
Step-by-step explanation: There are many less face cards in the deck than other cards, you would think youre getting 3 points each time rather than thinking 10 points each time
There is 1 2 of a pizza left for 4 friends to share. What fraction of a pizza will each friend get to eat? please help.
Answers
Answer:
1/3
Step-by-step explanation:
Number of pizza=12
Number of students=4
Therefore,4/12=4/4
=1/3
solve 4(x-7)+10=-10 A=X-2 B=X=4 C=X=1 D=X=3
Answers
Answer:
A. -2
Step-by-step explanation:
Find the vertex, axis of symmetry, and y-intercept of the quadratic function.
Answers
Answer:
See below ~
Step-by-step explanation:
Details of Graph
Maximum = 2Domain : all real numbersRange : y ≤ 2How do I solve 15=3p/6 ?
Answers
=> 15 x 6 = 3p
=> 90 = 3p
=> p =90/3
=> p = 30
Hope this helps you :)
-9v -6(8-7v)
Explain please
Answers
Answer:
33v - 48
Step-by-step explanation:
- 9v - 6(8 - 7v) ← multiply each term in the parenthesis by - 6
= - 9v - 48 + 42v ← collect like terms
= 33v - 48
[tex]\qquad\qquad\huge\underline{{\sf Answer}}[/tex]
Let's solve ~
[tex]\qquad \sf \dashrightarrow \: - 9v - 6(8 - 7v)[/tex]
[tex]\qquad \sf \dashrightarrow \: - 9v - [(6 \times 8) - (6 \times 7v)][/tex]
[tex]\qquad \sf \dashrightarrow \: - 9v - (48 - 42v)[/tex]
[tex]\qquad \sf \dashrightarrow \: - 9v - 48 + 42v[/tex]
[tex]\qquad \sf \dashrightarrow \:33v - 48[/tex]
Hope it helps ~
Find an equation of a line through (2, 1) and perpendicular to - 2x + 4y = 8.
y = -2x - 1
y = -2x + 5
О.
y = -2x - 3
O
y = -2x + 3
Answers
keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above
[tex]-2x+4y=8\implies 4y=2x+8\implies y=\cfrac{2x+8}{4} \\\\\\ y=\cfrac{2x}{4}+\cfrac{8}{4}\implies y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{1}{2}}x+2\impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}[/tex]
since we know that's its slope, then
[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{\cfrac{1}{2}} ~\hfill \stackrel{reciprocal}{\cfrac{2}{1}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{2}{1}\implies -2}}[/tex]
so then we're really looking for the equation of a line whose slope is -2 and passes through (2 , 1)
[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{1})\qquad \qquad \stackrel{slope}{m}\implies -2 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{1}=\stackrel{m}{-2}(x-\stackrel{x_1}{2}) \\\\\\ y-1=-2x-4\implies y=-2x-3[/tex]
Solve the equation 14x + 7y= 28 for x.
O A. x-y-2
OB. x = 28-7y
O C. x = 2-y
O D. x = 4-y
Answers
The equation that represents the solution of x is x = 2 - y/2
How to solve the equation?The equation is given as:
14x + 7y = 28
Divide through by 7
2x + y = 4
Subtract y from both sides
2x = 4 - y
Divide both sides by 2
x = 2 - y/2
Hence, the equation that represents the solution of x is x = 2 - y/2
Read more about equations at:
https://brainly.com/question/2972832
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Rearrange the formula below to make x the subject. α y+1=2x - 10/x+1
Answers
Answer:
cross multiplication
x
find the sum of n first natural numbers is 496 ,find the number of terms
Answers
Answer:
Number of terms = 31.
Step-by-step explanation:
Sum of n terms of an A S with first term 1 and common difference 1:-
496 = (n/2)(2(1) + 1(n - 1))
496 = (n/2)(n + 1)
n^2/2 + n/2 = 496 Multiply through by 2:-
n^2 + n = 992
n^2 + n - 992 = 0
(n - 31)(n + 32) = 0
n = 31 (we ignore the negative).
Pre-calc/trig
Suppose that θ=arcsin(1/7) in the first quadrant. What is cos(θ/2)? (round to 4 decimals)
Answers
Answer:
0.9974
Step-by-step explanation:
[tex]arcsin( \frac{1}{7} ) = 0.1433 \\ cos( \frac{0.1433}{2} ) = 0.9974[/tex]
the second choice is "reverse" and "do not reverse"
Answers
Answer:
Divide by -8reverseStep-by-step explanation:
Multiplying or dividing an inequality by a negative number requires that the inequality symbol be reversed.
__
To solve this inequality, we can divide by the coefficient of x, which is -8. Dividing by -8 means we need to reverse the inequality symbol.
_____
Additional comment
Actually, applying any function that has a negative slope means the ordering is reversed. That is the more general case.
Here, we're specifically concerned with division by a negative number. You can see the effect on the ordering relation by considering ...
-2 < 12 > -1 . . . . divide both sides by -1A right angled triangle has sides 12m and 5 m.what is the length of the hypotenuses of this triangle?
Answers
144 + 25 = c^2
169 = c^2
13=c
Answer:
13 m
Step-by-step explanation:
Given: sides of the right triangle are 12 m and 5 mTo find: Length of hypotenuseBy Pythagoras Theorem:[tex]l(hypotenuse)=\sqrt{12^2+5^2}[/tex][tex]\implies l(hypotenuse)=\sqrt{144+25}[/tex][tex]\implies l(hypotenuse)=\sqrt{169}[/tex][tex]\implies l(hypotenuse)=13\: m[/tex]what plus what equals -20
Answers
Here you are adding but since you have a negative 40, adding 20 (which is less than 40) the sum is still negative.
A concession stand owner bought some bottles of water for $0.75 each and began selling them for $1.25 each. When he made back all the money he spent, he still had 20 bottles left to sell. What equation can be written to determine the amount of money the owner spent?
Answers
Cácule o limite de:
lim [tex]3^{2n} + 1[/tex]
_____________________
n-∞[tex]4^{n+2} - 3[/tex]
R:∞
Answers
The limit of the expressions [tex]3^{2n} + 1[/tex] and [tex]4^{n + 2} - 3[/tex] for n to infinity is infinity
How to calculate the limits?The expressions are given as:
[tex]3^{2n} + 1[/tex]
[tex]4^{n + 2} - 3[/tex]
The limits of the expressions as they approach infinity are represented as:
[tex]\lim_{n \to \infty} 3^{2n} + 1[/tex] and [tex]\lim_{n \to \infty} 4^{n + 2} - 3[/tex]
Substitute [tex]\infty[/tex] for n in both expressions
[tex]\lim_{n \to \infty} 3^{2 * \infty} + 1[/tex] and [tex]\lim_{n \to \infty} 4^{\infty + 2} - 3[/tex]
Solve both expressions independently;
[tex]\lim_{n \to \infty} 3^{2n} + 1 = 3^{2 * \infty} + 1[/tex]
Evaluate the product
[tex]\lim_{n \to \infty} 3^{2n} + 1 = 3^{\infty} + 1[/tex]
Evaluate the exponent
[tex]\lim_{n \to \infty} 3^{2n} + 1 = \infty + 1[/tex]
Evaluate the sum
[tex]\lim_{n \to \infty} 3^{2n} + 1 = \infty[/tex]
Also, we have:
[tex]\lim_{n \to \infty} 4^{n + 2} - 3 = 4^{\infty + 2} - 3[/tex]
Evaluate the sum
[tex]\lim_{n \to \infty} 4^{n + 2} - 3 = 4^{\infty} - 3[/tex]
Evaluate the exponent
[tex]\lim_{n \to \infty} 4^{n + 2} - 3 =\infty - 3[/tex]
Evaluate the difference
[tex]\lim_{n \to \infty} 4^{n + 2} - 3 =\infty[/tex]
Hence, the limit of the expressions for n to infinity is infinity
Read more about limits at:
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Jackie has 3 brothers. Maria has 2 brothers. Keri has 1 brother. Harriet has 2 brothers. Between the four girls, what is the mean number of brothers? Median? Mode?
Answers
Answer:
mean is 2 brothers.
mean, means average
Someone show me step for step please!
Answers
[tex]\csc \theta \tan \theta - \tan \theta \sin \theta \\\\\\=\dfrac 1{\sin \theta} \cdot \dfrac{\sin \theta }{\cos \theta }- \dfrac{\sin \theta }{\cos \theta} \cdot \sin \theta\\\\\\=\dfrac{1}{\cos \theta}-\dfrac{\sin^2 \theta}{\cos \theta}\\\\\\=\dfrac{1- \sin^2 \theta}{\cos \theta}\\\\\\=\dfrac{\cos^2 \theta}{\cos \theta}~~~~~~~~~~~~~~;[\sin^2 \theta + \cos^2 \theta =1 ]\\\\\\=\cos \theta[/tex]
Let's write out some trigonometric identities that we can use:
[tex]tan(x)=\frac{sin(x)}{cos(x)} \\[/tex][tex]csc(x)=\frac{1}{sin(x)}[/tex][tex]sin^2(x)+cos^2(x)=1[/tex] ⇒ [tex]cos^2(x)=1-sin^2(x)[/tex]Now let's try solving them out:
[tex]csc(x)tan(x)-tan(x)sin(x)=\frac{1}{sin(x)}tan(x)-sin(x)tan(x)\\\\ csc(x)tan(x)-tan(x)sin(x)=tan(x)(\frac{1}{sin(x)}-sin(x))\\\\ csc(x)tan(x)-tan(x)sin(x)=tan(x)(\frac{1}{sin(x)} -\frac{sin^2(x)}{sin(x)} )\\\\csc(x)tan(x)-tan(x)sin(x)=tan(x)*(\frac{1-sin^2(x)}{sin(x)} )\\\\csc(x)tan(x)-tan(x)sin(x)=\frac{sin(x)}{cos(x)}*\frac{cos^2(x)}{sin(x)} \\\\csc(x)tan(x)-tan(x)sin(x)=\frac{sin(x)}{sin(x)}*\frac{cos^2(x)}{cos(x)}\\\\csc(x)tan(x)-tan(x)sin(x)=1 * cos(x)\\\\ csc(x)tan(x)-tan(x)sin(x)=cos(x)[/tex]
*I used a different variable, but that doesn't change the answer
Answer: cos(x)
Hope that helps!
To graph the inequality x > -3, draw an open circle on -3 and shade to the left.
TrueFalse
Answers
Annie bought a package of 100 pens for $167.00. How much did each
pen cost?
Answers
Answer:
Step-by-step explanation:
167.00 / 100 = $1.67