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Find the value of the number $ a $ such that the families of curves $ y = (x +c)^{-1} $ and $ y = a(x + k)^{1/3} $ are orthogonal trajectories.

$a=3^{1 / 3}=\sqrt[3]{3}$

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in this problem were given to families off chairs and where supply body of a size that peace stickers are targeting. So we're gonna use the first, um, family were going to say that X plus C is for war. And from this we see that see isn't the corner. What? When it's X? Now let's take herto both sides. Suspect X. We have negative boy squared. That's why prime minus one is zero. Since he just constant and promise we see that weapon is an equal negative war. Square it to terms or to punching. Our target will teach. You know that smokes multiplication of the slopes, his negative one. So if this is and one minute to slope off this one's into, we know that M one times and two should be negative on. So if you can find the slope of entries than negative owner and one where everyone is nature wife's credit for that it's negative over negative, widespread. So the slope should be Walk over White spirit support a 2nd 1 w have you are D X. Is it with Warner once? Craig. So this means that you are convoy spread is the DX are defeated Interval off both sides. You see that? Like you plus three should be called X possum Integration conference C and what you is then three times exposed. Okay. Now, why is the Nico to three 1 30 plus experts K to the problem on third now compared dysfunction this family occur which the given one. We have a times Expos K 1/3. We want those to be the same. Okay. And these parts are the same. Such means that they should be with three power one.