Help me guys!
Show that the locus of the middle points of normal chords of the parabola y² = 4ax is
y⁴ - 2a(x - 2a)y² + 8a⁴=0
Chord of Contact:-
The chord joining the points of contact of two tangents drawn from an external point to a parabola is known as the chord of contact of tangents drawn from external point.Equation of the normal chord at any point (at², 2at) of the parabola y² = 4ax is
y + tx = 2at + at³ ....(i)
Look at the attached figure
But if M (x₁, y₁) be it's middle point its equation must be also,
T = S₁
:⟹ yy₁ - 2a (x + x₁) = y₁² - 4ax₁
:⟹ yy₁ - 2ax = y₁² - 2ax₁ .....(ii)
Therefore, from eqs. (i) and (ii) are identical, comparing, them
[tex] \sf \frac{1}{y_1} = \frac{t}{ - 2a} = \frac{2at + {at}^{3} }{ {y_1}^{2} - 2ax} \\ \\ \sf \: from \: first \: two \: relations \: ,t = - \frac{2a}{y_1} ....(iii)\\ \\ \sf \: from \: first \: two \: relations \: , \: \frac{t}{ - 2a} = \frac{2at + {at}^{3} }{{y_1}^{2} - 2ax_1} [/tex]
[tex] \implies \sf \: \frac{{y_1}^{2} - 2ax_1}{ - 2a} = 2a + {at}^{2} \: \\ \\ \implies \sf \: \frac{{y_1}^{2} - 2ax_1}{ - 2a} = 2a + a \bigg \lgroup \frac{ - 2a}{y_1} { \bigg \rgroup}^{2} \qquad\{ \: from \: eqs. \: (iii) \} \\ \\ \implies \sf \:\frac{{y_1}^{2} - 2ax_1}{ - 2a} = \frac{2a{y_1}^{2} + 4 {a}^{3} }{{y_1}^{2}} \\ \\ \implies \sf \: {y_1}^{4} - 2ax_1 {y_1}^{2} = - 4 {a}^{2} {y_1}^{2} - 8 {a}^{4} \\ \\ \implies \sf \: {y_1}^{4} - 2a(x_1 - 2a){y_1}^{2} + 8 {a}^{4} = 0[/tex]
[tex] \sf hence, \: the \: locus \: of \: middle \: point \: (x_1,y_1) \: is \: \\ \\ \qquad \qquad \: \sf \: {y_1}^{4} - 2a(x_1 - 2a){y_1}^{2} + 8 {a}^{4} = 0[/tex]
The ratio of the number of peanuts to the number of cashews in a trail mix recipe is 5:3.
Complete the table to show the relationship between peanuts and cashews in the trail mix recipe. Enter a number in each box.
Answer:
40 and 16 for 8 servings
25 and 10 for 5 servings
15 and 6 for 3 servings
Step-by-step explanation:
i need help with this, graphing
Answer:
30 baloons per hourStep-by-step explanation:
Use some points on the graph:
(0, 0), (2, 60), (4, 120), (8, 240)The graph is a line passing through the origin, so this is a proportional relationship.
Standard form of such relationship is
y = kx, where k- coefficient of proportionalityUsing one of the coordinate pairs find the value of k
60 = 2kk = 60/2k = 30This is indicating the number of baloons filled per hour is 30.
Simon wants to buy some flowers He goes into a flower shop and buys 7 daffodils and 6 daisies Simon wants to know the cost of daffodils per flower Unfortunately, Simon forgot to get a receipt, however, he can remember that daisies costs 61p per flower. He can also remember that he spent £8.49 What is the cost of daffodils per flower?
Answer:
69p each
Step-by-step explanation:
well if each daisy is 61p you want to multiply that price by 6 to get total daisy price, which is £3.66 Then you take £8.49 from £3.66 to get the total price of the daffodils, which is £4.83. Then you divide the the price by the number of daffodils which is 69p. :))(
Jerry is mowing his lawn.It takes him 1/6 of an hour to mow 3/20 of his yard. What proportion of the yard does jerry mow per hour
Answer:
If he mows 3/20 in 1/6 of an hour he'll mow 18/20 or 9/10 in one hour
which means the ratios are equal
1/6:3/20 = 1:18/20 = 1:9/10
Therefore, Jerry will mow 90% of the lawn in one hour
Step-by-step explanation:
Hope this helps:)...if not then sorry for wasting your time and may GOd bless you
!!!!!help please!!!!!
to G H I would be 2x2x3.
The tallest of a collection of wooden dolls stands 1 meter tall, and each successive doll is half as tall as the one before it. If there are 8 dolls in the collection, which is closest to the combined height of the dolls?
A. 1.5 meters
B. 2 meters
c. 3.1 meters
D. 4 meters
Answer:
the answer is a for approval
Mike has $21 to spend at the mall.
He spends all of his money on
bracelets for his sisters. Bracelets
cost $3 each. How many bracelets
does he buy?
Answer:
7 Bracelets
Step-by-step explanation:
Okay, you have 21$ to spend, since bracelets cost 3$ each (not including tax), you can do 21/3 which will get you 7 bracelets. Hope this helps!
Answer:
7 bracelets
Step-by-step explanation:
21/3 = 7
9×100+9×10+0×1+0×1/10+3×1/100+7×1/1000
Answer: 990.037
Step-by-step explanation:
About 14,200,000 results (0.72 seconds)
990.037 +9×10+0×1+0×1/10+3×1/100+7×1/1000
which expression is equivalent to 5^4 × (5^-3)^2?
Answer:
5^-2 or 1/25
Step-by-step explanation:
use index laws.
1. (A^m)^n = a^mn
5^4 x (5^-3)^2=
5^4 x 5^-6
2. A^m x A^n = A^m+n
5^4 x 5^-6
5^-2=
1/5²=
1/25
Answer:
[tex]5^{4}\times \left( 5^{-3}\right)^{2} =5^{-2}[/tex]
Step-by-step explanation:
[tex]5^{4}\times \left( 5^{-3}\right)^{2} =5^{4}\times 5^{-6}=5^{4-6}=5^{-2}[/tex]
We can write:
[tex]5^{-2}=\frac{1}{5^{2}} =\frac{1}{25}[/tex]
What is the third angle on a triangle with angels of 45 and 45?
[tex] \huge\fbox\orange{ANSWER} [/tex]
[tex] \huge\mathsf \pink{90°}[/tex]
X + 45° + 45° = 180° X + 90° = 180°X + 90° - 90° = 180° - 90° X = 90°x over 4 space equals space 6 over 8 ; x=
Answer: x=3
Step-by-step explanation: This is a porportion, meaning that the numbers are set equal to each other. Look at your denominators, 4 and 8? What do they have in common? In this case, they have a factor of 2 in common, and 8 is 4x2. So then we look at the numerators now. What times 2 equals to 6? Your answer then becomes 3.
(-3)+5+(-5)+12=
Can someone break this down for me?
I just wanted to get some help ...
Thank you !
Answer:
[tex]9[/tex]
Step-by-step explanation:
[tex](-3)+5+(-5)+12=\\-3+5-5+12=\\5-5-3+12=\\-3+12=\\12-3=\\9[/tex]
Whenever you are doing a problem that says [tex]+(-n)[/tex] (n for number), it's the same as subtracting that number.
The diagram shows the locations of the sun, Earth, and the moon. Which of these is possible only when the sun, Earth, and the moon are aligned as shown?
Answer:
It will experience a low tide because water is drawn away from the area between the high tides.
Explanation:
The oceans on earth bulge outwards towards the sun and moon due to gravitational pull. This building is highest when the moon and sun are in line (same plane). The coastline within this plane will experience high tides while those at the right angle will have the lowest tides.
If 65 is 20% of x, then x = ?
Answer:
325
Step-by-step explanation:
I got it right before
Answer:
325
Step-by-step explanation:
Question 15.
Which expression is equivalent to
(4x³ + 5x² + 2) + (3x² - 7x + 4)?
Answer:
4x³ + 8x² - 7x + 6
Step-by-step explanation:
remove the brackets by distributing the sign.
4x³ + 5x² + 2 + 3x² - 7x + 4
4x³ + 8x² - 7x + 6
Gather like terms.
If you want to factorise (optional)
4x³ + 8x² - 7x + 6 =
4x²(1 + 2x) - 6x + 3 + 3 - x
4x²(1 + 2x) - 3(1 + 2x) + 9 - x
(4x² - 3) (1 + 2x) + 9 - x
The circumference of a wheel is 302 centimetres. Determine the diameter of the wheel. Round your answer to the nearest tenth.
At Freddy’s fro yo shop 25 percent of the 44 frozen yogurt flavors are sugar free how many frozen yogurt flavors are sugar free
What slope would make the lines
perpendicular?
y = -1/3x +5
y = [?]x+2
13) Which value is equivalent to the expression 7^2 - 3^4
A) -32
B)-19
C) 16
D) 32
Answer:
-32
Step-by-step explanation:
7^2 = 49
3^4 = 81
49-81=
-32
what is the cost of 23.5 gallons of gas
Answer:
Too much
Step-by-step explanation:
I did the math myself
can someone help? Please :(
jamie read an 80 page book in 2 hrs. how long would it take him to read a 100 page book?
Answer:
1.5 hours
Step-by-step explanation:
80 /120= 100/x
80x=12000
12000/80=150
150 minutes- 1.5 hours
Answer:
Around 2 hours and 50 minutes.
Solve
If Jamie can read an 80 page book in 2 hours, divide 100 by the time, 2 hours.
[tex]100 / 2 = 50[/tex]
Add that to the time, 2 hours.
[tex]= 2~hrs.~50~mins.[/tex]
Which expression is equivalent to m^2 - m - 12 ?
a) (m + 2)(m - 6)
b) (m - 3)(m + 4)
c) (m - 2)(m + 6)
d) (m + 3)(m - 4)
need help will give brainlist
Answer:
3rd option
Step-by-step explanation:
- 1 ( [tex]\frac{3x}{2y}[/tex] )
- 1 is multiplying the complere fraction , not just 1 part of it , so
= - [tex]\frac{3x}{2y}[/tex]
Find the equation of the line shown
Answer:
the equation of the line is 2y-x-1
pls give brainliest
help!! Graph y>−5/4x+5
Answer:
it's the top right.
Step-by-step explanation:
hope this helps.
I. GIVEN TRIANGLE POR, DETERMINE THE FOLLOWING.
please help me out with this
34
Step-by-step explanation:
122
This is a lengthy question ill go through all the formulas
This is a right triangle which means u can use pythagorean theorem to find QP
formula :
[tex]a^{2} +b^{2} =c^{2}[/tex]
Note that [tex]c^{2}[/tex] is always the hypotenuse(the hyp is always opposite to ninety degrees)
so lets substitute
=> [tex]a^{2} +17^{2} =28^{2}[/tex]
=> [tex]a^{2} +289 =784[/tex]
subtract 289 both sides
[tex]= > a^{2}=495[/tex]
Now square root both
[tex]\sqrt{ a}^{2}=\sqrt{495}[/tex]
==> [tex]3\sqrt{55}[/tex] or 22.2 (answer)
But for the other three u have to use another formula any of sin, tan,cos while using For example angle Q and then use any and do inverse it will be very easy after u know all the formulas of Sin,cos, tan
SOHCAHTOA
^
Thats an acronym for the formula
Sin= Opposite/ hyp
Cos= Adjustent/Hyp
Tan= Opposite/Adjustent
It takes Don 55 minutes to finish his
math homework and 25 minutes to
finish his reading homework. How
many more minutes did Don spend
on his math homework than on his
reading homework?
Answer:
Don spent 30 more minutes on his math homework than reading homework.
Step-by-step explanation:
55 - 25 = 30
HELP PLEASE! The perimeter is 22. How much bigger is the longest side than the shortest side?
I asked this question two hours ago, and since then, I haven't recived any help. I don't mean to sound demanding and I understand that there are millions of users here, but it seems it's been a while since someone's answered this, and I can't figure it out.
Answer:
[tex]\frac{52}{7}[/tex] [tex]units[/tex]
Step-by-step explanation:
Add up all the sides.
(2a - 3) + (3a + 1) + 2a = 7a - 2
Lets set it equal to 22 since it is the perimeter
7a - 2 = 22
add 2 both sides
7a = 24
a = [tex]\frac{24}{7}[/tex]
From the image, you can tell that 2a - 3 is the shortest side and 3a + 1 is the longest side.
(3a + 1) - (2a - 3)
Substitute [tex]\frac{24}{7}[/tex] in for a.
You should be able to get [tex]\frac{52}{7}[/tex] units.