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Show that the sum of the $ x- $ and $ y- $ interc…

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Problem 45 Hard Difficulty

Find an equation of the tangent line to the hyperbola
$ \frac {x^2}{a^2} - \frac {y^2}{b^2} = 1 $
at the point $ (x_o, y_o). $

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Video Transcript

In this problem, we are given an equation and were asked to find an equation of potential line for line or not. We know that the question of the line will be for y minus t is equal to the slope or the derivatur of this function. It will be a given point of p by x minus x. Not so we first need to torquay the derivative at this given point. So, let's differentiate this given equation, we have 2 x over a square minus 2 y over b squared times y prime is equal to 0 point from this. Then we see that y prime is equal to x over a square times v squared over y, or we can write this 1. As if you were to relate the given point, we would find y prime to be equal to b squared over a squared mltiplied by x, not over y, not now. If that is why primeve point, let's right, equal structor, tangent line again, we have 1 minus 1. Is equal to b squared over a squared times x, not over? Why not multiplied by x minus x? Not now, let's multiply both sides by a square times y, not we would end up with a square times y, not times y minus a square times y. Not squared, is equal to b squared x, not times x, minus b, squared x, not square. We can write further this 1 as a square y, not times y minus b, squared x, not times x, that is equal to v square that is equal to b squared x, not squared minus 8 squad y, not squared all right. Well, that should be the other way around. This should be a square y, not squared minus b squared x, not square all right now, if we divide both sides by a square times b squared, he would end up with x, not or y, not times y divided by b, squared minus x, not times x, Divided by a square is equal to y, not squared, divided by b square minus x, not squared divided by a all right. This is same as x, not times x. Over a squared minus y, not times y over b squared, is equal to x, not squared over a square plus or minus y, not squared over square. So basically, i multiply both sides by negative 1 point. Now, look at the right hand, side here and look at the original equation. If you plugged given point x and a 1 into the original question, it would end up with 1 and since right hand. Side is the same. We'Re going to say that right hand. Side is equal to 1 side of the equation of the tangent line is x not times x, divided by a square minus y, not times y divided by b. Squared is equal to 1.

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Video Transcript

James Stewart

Calculus: Early Transcendentals

Grace He

Catherine Ross

Caleb Elmore

Kristen Karbon

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

Calculus: Early Transcendentals

Grace He

Catherine Ross

Caleb Elmore

Kristen Karbon

James StewartCalculus: Early Transcendentals

Grace He

Catherine Ross

Caleb Elmore

Kristen Karbon

James Stewart

Calculus: Early Transcendentals