Problem

Find an equation of the tangent line to the hyper…

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Problem 44 Hard Difficulty

Show by implicit differentiation that the ellipse
$ \frac {x^2}{a^2} + \frac {y^2}{b^2} = 1 $
at the point $ ( x_o, y_o) $ is
$ \frac {x_o x}{a^2} + \frac {y_o y}{b^2} = 1 $

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Video Transcript

In this problem, we are given an equation of an aeroplane where, as to show that, at point x, not 1 equation of the tangent line. Is this given equation all right? We know that equation of the tangent line is of the form y minus y. Not is derivative of this function. It will let that given point times x minus x. Not so we need to find a derivative of this function. Lets differentiate the given equation. We have 2 x over a square plus 2 over v. Squared times y prime is equal to 1. Now, let's plug the given point in we have 2 times x, not over a squared plus 2 times y, not over b squared times y prime is equal to it, while this should be 0 since it's a constant. This is 0 from this time we see that y prime is equal to negative b squared over a square times x, not over y, not all right now, if that is y prime, we can rewrite equation of tangent lie. Is y minus y not is equal to negative b squared over a square times x, not over y, not multiplied by x, ponent xro. Now, let's divide both. Let'S multiply both sides by a square times y spread. This would give us a square y, not times y minus a square y, not squared, is equal to negative b squared x, not times x, plus, b squared times x, not square all right now we can write this 1 as a squared times y, not times y Plus b squared times x, not times x, is equal to a square y in s square plus b, squared x, not squared. Now, let's multiply both sides or, let's divide both sides by 1 over a square times b square b would end up with y, not times y. Over b, squared plus x 9 times x, divided by a square is equal to. Why not spread over b, squared plus x, not squared over a square, now, look at the right hand, side. We have here and look at the original equation. If you put x not and y not into this equation, we would get the right hand side right. So we know that this right hand side is equal to 1. So then we end up with y, not over y of b squared plus x, not times x, over square is equal to 1 point and, as you can see, that is actually the equation of the tangent line.

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James Stewart

Calculus: Early Transcendentals

Catherine Ross

Kristen Karbon

Samuel Hannah

Joseph Lentino

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus: Early Transcendentals

Catherine Ross

Kristen Karbon

Samuel Hannah

Joseph Lentino

James StewartCalculus: Early Transcendentals

Catherine Ross

Kristen Karbon

Samuel Hannah

Joseph Lentino

James Stewart

Calculus: Early Transcendentals