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Thus question is related to a specific problem. I don't know how kindly you take to that.

Anyways, given this curve

$ y^2 x + a = x^2 + y^2 $

And this tangent

$ y = \frac{3}{2} x - 2 $

find $a$.

I've tried lot's of things. First I calculated the derivative of the curve, then I figured it must be equal to $3/2$. Then, in the resulting equation, I expressed $x$ using $y$ and tried to insert that $x$ both into the curve, and the tangent. Neither of those attempts lead to anything. I always get some ugly numbers that don't appear correct when I input them into wolfram alpha.

What do I do?

Luka Horvat
  • 2,618

1 Answers1

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Let the point of tangency be $(p,q)$.

By implicit differentiation, we have $y^2+2xyy' =2x+2yy'$.

Since the slope at $(p,q)$ is $3/2$, we get $$q^2+3pq=2p+3q.\tag{$1$}$$ Also, since $(p,q)$ is on the tangent line, we have $$q=\frac{3}{2}p-2.\tag{$2$}$$ Use $(2)$ to substitute for $p$ in $(1)$. We get the quadratic equation $9q^2-q-8=0$. The quadratic even factors nicely. Thank you, problem setter!

Finally, substitute for $p$ and $q$ in the equation of the curve to find $a$.

André Nicolas
  • 507,029
  • Hey! That was way simpler than I thought it would be. Thanks a ton. Btw., does it make a difference if we use (2) to substitute for q in (1)? – Luka Horvat Apr 07 '13 at 10:45
  • You are welcome. Always good to hear that the problem is now viewed as simple. As to the variable one substitutes for, it makes no difference, but I think you will find my choice takes a few seconds less. – André Nicolas Apr 07 '13 at 17:09