In Problems 92-98, use Descartes's method from Problem 91 to find the equation of the line tange

In Problems 92-98, use Descartes's method from Problem 91 to...

Key Concepts

Derivative

A measure of how a function changes as its input changes. When finding the tangent line to a curve, the derivative at the point of tangency gives the slope of that tangent line.

Application of Descartes's Method

A systematic procedure for determining the equation of the tangent line to a given curve. It involves differentiating the equation (often implicitly) and then using the point-slope form of a line to form the equation of the tangent line.

Implicit Differentiation

A technique used for finding the derivative of functions defined implicitly rather than explicitly. This method involves differentiating both sides of an equation with respect to the independent variable while applying the chain rule to any term involving the dependent variable.

Tangent Line

The line that just touches a curve at a given point and has the same slope (derivative) as the curve at that point. It is used to approximate the curve near the point of tangency.

Transcript

00:01 So we need to give this cardage method to find the tangent line to 2x square plus 3 y square equals to 14 at 1 comma 2 so let's assume a tangent line is y equals to mx plus b now this point satisfy this equation therefore 2 is equal to 1 times m plus b and therefore we get b as 2 minus m so now our system of equation is first one is given as it is 2x squared plus 3y squared equals to 14 and second one one is our assumption which is y equals to mx plus b but b is 2 minus m great now this equation should satisfy this since they touch at a point so therefore 2x squared plus 3 times mx plus 2 minus m whole square is equal to 14 so on squaring this we get 3 times so applying this we get 3 times so applying a plus b whole square formula so mx squared plus two times mx plus two minus m plus two minus m square minus 40 is equals to zero correct so we get here two x square multiply three inside so we get three m square x square here we get multiplying two m inside so we have four m minus two m square x and here again we have to apply whole square formula okay by multiply this three will get multiplied here as well and with this so 4 minus 4 m plus m square minus 14 is equal to 0 so on combining this we get 2 plus 3 m square x square plus multiplying 3 inside gives us 12m minus 6 m square x and if you multiply this 3 inside and take this 14 so this is 12 minus 14 which is minus 2 then 3 minus 4m is minus 12m and 3 times m square is 3m square and this is equals to 0 okay now this equation to give a unique solution you must have a discriminator 0 therefore b square minus 4 ac should be 0 and therefore 2 plus 3 m square whole square minus 4 times a sorry that's not a b so b is this so 12 m minus 6 m square whole square minus 4 times a which is 2 plus 3 m square times c which is 3 m square minus 12m minus 2 should be equal to 0 so on squaring this we get 144 m square plus 36 m raised 2 minus 144 m q and here minus 4 times if we multiply 2, so we get 6m square, then 2 times minus 12m is minus 24m, 2 times minus 2 times minus 2 is minus 4.

03:18 Then we have to multiply by 3m square, so 3m square into 3m square, 9m square, then 3m square into minus 12m, so minus 36m cube, and then 3m square minus 2 is minus 6m square.

03:36 Right this is equal to 0 okay so in next step we can multiply four inside so we have 144 m square 36 m raised to 4 minus 144 m cube minus 24 m square minus minus plus 24 into 4 is 96 m square minus minus 4 into minus 4 plus 16 minus 4 into plus 9 is minus 36 m square minus 4 into minus 36 is plus 6m square minus 4 into minus 36 is plus 144 m cube then minus 4 into minus 6 is plus 24 m squared okay so that is a term so now let's see if we can cancel some terms so here we have minus sorry plus 36 times m raised to 4 and here we have minus 36 m raised 2 sorry this must be m raised to 4 right so this get cancel then n q so minus 144 mq plus 144 m cube then in rest of the terms we can add right so 144 m square minus 24 m square plus 24 m square so this get cancelled so you have left with 144 m square next term is m so yeah we have only one m that is 96 m and constant is 16 which is equal to 0 so on solving this for m we get m is equal to minus 1 by therefore, our equation is y equals to minus 1 by 3x plus 2 minus 1 by 3...

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