please help with calc!

[orchid crown]

the problem is:

Show by implicit differentiation that the tangent to the ellipse x^2/a^2 + y^2/b^2 = 1 at the point (x_0, y_0) is x_0 x/a^2 + y_0y/b^2 = 1

i apologize for the horrible notation—i attached a photo if that helps

[lapis arrow]

so which oart are you facing problem in

find slope using implicit differentiation

[orchid crown]

i understand how to differntiate it and everything, but i'm stuck making it look like the final thing. after i take the derivitive of the ellipse's equation and plug in x0 and y0 into my slope, then put it in point-slope form, i can't simplify it

[lapis arrow]

then use point slope

you get this

[orchid crown]

well like yes

but no 😭

because i still can't simplify it properly

[lapis arrow]

yes?

[orchid crown]

ah you separated them

ok i had them as a common denominator

[lapis arrow]

you manipulate it to this form

then observe that x0 and y0 lie on the ellipse

so it should satisfy the equation

[orchid crown]

thank you!

[lapis arrow]

hence u can write rhs =1

[orchid crown]

sorry for all that trouble just for the issue to be i didn't separate the fration

[lapis arrow]

and you got your answer

[lapis arrow]

no problem