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Problem 47 Hard Difficulty

Show, using implicit differentiation, that any tangent line at a point $ P $ to a circle $ O $ is perpendicular to the radius $ OP. $

Answer

$y^{\prime}=-\frac{x}{y}$, $P\left(x_{0}, y_{0}\right)$ is $-\frac{x_{0}}{y_{0}}$, $\frac{-1}{-x_{0} / y_{0}}=\frac{y_{0}}{x_{0}}$

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Video Transcript

in this problem where has to show that radius Circle Salt Line LP is perpendicular to tension Line Dad Pass. It's true. Point P Solicitor, Central city of circles Looking back this year. Zero. So that's origin and 20 is that X Not? Why not? All right, being under the question of the circle is expert plus Waas for is our spirit. And let's say that we're interested in the question tangent line till this. First of all, it's different ship. This question, which is actually have to was, too. Why times y prime is equal to zero. So from this we see that twos will cancel out. And what kind would then be negative x over? Why now that the question of kinds of line we know it is a Hawaiian islands for not visiting the white crime ex, not times Next month's excellent We know what oi primary is. So let's just play everything you know that point peace an excellent one. So we have one minus or not the secret tomb. Now they would take a bullet at a given point booth and be negative off X, not p Y y. No terms export six knots so This is the question of potential. All right, Now let's find any question for radio. So line, Opie line. Oh, Pete, No, That'll be all for why, minus zero. We're pulling this right here again. So why minus zero is equal to We need to find the slope. But we know the two points that that line passes troops. So I gotta be wine on minus zero. Do y x not my serum. That is slow. Well liked by X minus zero. So if that is the slope off line of P and this part is the slope, I'll change it. Lane, if we were to multiply those two So negative x not over. Why not? Don't spite by wine. Excellent. We'll see that multiplication up the slope skates us negative on. So this means that those two lines are beach perpendicular.