Implicit derivation

[rancid fulcrum]

I found the derivative of the function, but idk what to do with it.

[sage wolf]

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[modern latch]

The idea is simple

Is to find the coordinate of the intersection point between both tangent line to the ellipse

so the y-coordinate of the intersection point is x and the x-coordinate of the intersection point is already given which is 3

The derivative that you have find is the slope of both tangent lines

You will a system of equations

will find

The equation of the first tangent line will be

y=0.25(x+5)

and the second equation is

sorry dont need to find the equation of the second tangent because there is no more information but since both tangent line intersect at the same then the length of the lampada is the value of y for x=3

so the required value of x is 0.25(3+5)=2

2 units of length measure

[rancid fulcrum]

How do I discover this?

[rancid fulcrum]

you were talking about the line (1)?

[modern latch]

2

[rancid fulcrum]

Ok

[modern latch]

y=0.25(x+5)

[rancid fulcrum]

How?

[modern latch]

line 1 you cannot find his equation

[rancid fulcrum]

Ok

[modern latch]

look

take (a,b) is the point of tangency of line 1 to the ellipse

[rancid fulcrum]

$\frac{dy}{dx} = -\frac{x}{4y}$

[granite heath]

neruguis

[modern latch]

good

the equation of the tangent line is y=mx+b

[rancid fulcrum]

I thought in a triangle

[modern latch]

where m=-x/4y at (a,b)

no listen

[rancid fulcrum]

a/b is the derivative

[rancid fulcrum]

Ok

[modern latch]

the slope of the line will be m=-a/4b

[rancid fulcrum]

yes

but a is the length

And b height

[modern latch]

a and b are the coordinates of the point of tangency

bro

[rancid fulcrum]

and -5 - a

is the distance to 0

[modern latch]

dont make resoning by distances

[rancid fulcrum]

$x^2 + 4y^2 = 5$

[granite heath]

neruguis

[modern latch]

you will be lost

[rancid fulcrum]

then x is -5-a

$(5+a)^2 + 4b^2 = 5$

[modern latch]

no the equation of line 1 will be y=(-a/4b)x+b so to find b use the point (-5,0)

after plug the point (a,b) into the equation of the ellipse

[rancid fulcrum]

isn’t it line 2

[granite heath]

neruguis

[modern latch]

then solve the system of equations y=(-a/4b)(x+5) and

[rancid fulcrum]

this is true, right?

[modern latch]

the abobe equation

[rancid fulcrum]

I’m trying

ok thanks

[modern latch]

no a^{2}+4b^{2}=5

bro listen

[rancid fulcrum]

ok

[modern latch]

why you have used 5+a

[rancid fulcrum]

-5-a

[modern latch]

it should only be a

[rancid fulcrum]

yes

[modern latch]

y=(-a/4b)(x+5) and a^{2}+4b^{2}=5

solve those equations

to find the value of a and b

[rancid fulcrum]

it’s -5+a

b = y

-5+a = x

(x,y) is the point of tangency

[modern latch]

it will not help you

[rancid fulcrum]

ok

[modern latch]

since it is calculus

[rancid fulcrum]

it’s still so much information needed

[modern latch]

you must find the equation of the tangent line 1

[rancid fulcrum]

we just have -5 and 3 of numbers

And the equation

[modern latch]

why you dont listen

the aim of the task is to find the equation of line 1

[rancid fulcrum]

really?

[modern latch]

because you can plug the value x=3 into this equation to find the height of the lampada

yes bro

[rancid fulcrum]

wait

yes

but the only thing we can fix is the line 2 passing on (-5,0) and touching the ellipse

[modern latch]

line 1 not 2

following ur draw

[rancid fulcrum]

if the height of the lamp change, we still wouldn’t know

[modern latch]

the x-coordinate of the lampada is fixed

[rancid fulcrum]

I think we can find the tangent line just by knowing that

[rancid fulcrum]

this

but idk how

[modern latch]

the height of the lampda is fixed because the light must be tangent to the ellipse

any way the height is 2

[rancid fulcrum]

well yes

[modern latch]

think about it

[rancid fulcrum]

I found 4 in Desmos

the inclination

1/4

then the height is 2

brb

bye

[modern latch]

correct

hahahahahahahah

[rancid fulcrum]

but I didn’t use calc

lol

[modern latch]

use calculus you will arrive to the same answer

[rancid fulcrum]

I’ll try later

Bye

And thanks for all ur time

[modern latch]

ur welcome

[rancid fulcrum]

I’m trying to go to the answer to the problem

Then find the inverse path

[modern latch]

you want to solve problem by distance

[rancid fulcrum]

And try to find the coords of the tangency point

[modern latch]

you cannot because the hypotenuse is not known

[rancid fulcrum]

It’s a good problem

[rancid fulcrum]

no?

[modern latch]

yes

[rancid fulcrum]

Ok

[modern latch]

if you wanna any help in calculus

I'm here

[restive locust]

true so how do we find the specific answer (im sorry i skipped most of the convo, so idk if this topic has been covered)
the can calc tangent 2 but it can be any point lying on that line right?

[modern latch]

Hii

Look the aim of the task is to find the y-coordinate of the point which has 3 as x-coordinate

using any tangent line

For the tangent line (right in pic) we dont have any info about a point that this line passes through

but for line 1(left in pic) yes which is (-5,0)

so its easy to use that tangent line to find the length of the lampada

[rancid fulcrum]

I solved

[modern latch]

b is already known

b=3-(-5)=8

if we follow ur solution

your solution is not correct

bro

you want see the solution with calculus

It is not jsutified why have you used the value of x=-1 to find a point on a the tangent line

why not choose x=-3

hhhh u see

since you found that (-1,1) is the point of tangency then find the equation of the tangent line directly using the points (-5,0) and (-1,1)

and plug the value of x=3 into that equation to find the value of y

[rancid fulcrum]

yes

the inclination is 1/4

then plug 5+3

8/4 = 2

[rancid fulcrum]

oh I did

+close