Math Help - Intersection of the Slopes of Curves

1. Intersection of the Slopes of Curves

If m is the slope of the curve $xy=3$ and m2 is the slope of the curve $x^2-y^2=4$, then at a point of intersection of the two curves:

1. (A) m1 = m2
2. (B) m1= -m2
3. (C) m1*m2 = 1
4. (D) m1*m2 = -1
5. (E) m1*m2 = -2

I think I might be doing it wrong... >_<

First, I found the derivative of the first one explicitly.
$
\frac{dy}{dx}[3x^-1]
$

which is:

$
\frac{-3}{x^2}
$

And then, the derivative of the second one implicitly:

$
\frac{dy}{dx}[x^2-y^2]=\frac{dy}{dx} [4]$

which is:

$
\frac{dy}{dx}= \frac{x}{y}$

But I can't see a relationship between the two slopes...?

2. Originally Posted by StarlitxSunshine
If m is the slope of the curve $xy=3$ and m2 is the slope of the curve $x^2-y^2=4$, then at a point of intersection of the two curves:

1. (A) m1 = m2
2. (B) m1= -m2
3. (C) m1*m2 = 1
4. (D) m1*m2 = -1
5. (E) m1*m2 = -2

I think I might be doing it wrong... >_<

First, I found the derivative of the first one explicitly.
$
\frac{dy}{dx}[3x^-1]
$

which is:

$
\frac{-3}{x^2}
$

And then, the derivative of the second one implicitly:

$
\frac{dy}{dx}[x^2-y^2]=\frac{dy}{dx} [4]$

which is:

$
\frac{dy}{dx}= \frac{x}{y}$

But I can't see a relationship between the two slopes...?

Multiply both slopes...

Tonio

3. O.o I tried that...

$
\frac{-3}{x^2} * \frac{x}{y} = \frac{-3}{xy}
$

That doesn't help... ? -3 isn't a choice in the answers...

4. You need to consider the slopes "at the point of intersection". So you need to find that point first, evaluate both gradients (at that point) then look for the answer.

5. Originally Posted by Debsta
You need to consider the slopes "at the point of intersection". So you need to find that point first, evaluate both gradients (at that point) then look for the answer.
Ohh...Umm.. :33

To find the point of intersection, should I make the table of values for each equation ? Or is there another way ?

6. Sub y = 3/x from first eqtn into the second one and do it algebraically.

7. Originally Posted by StarlitxSunshine
O.o I tried that...

$
\frac{-3}{x^2} * \frac{x}{y} = \frac{-3}{xy}
$

That doesn't help... ? -3 isn't a choice in the answers...

But you're given what xy is, girl!

Tonio

8. Originally Posted by StarlitxSunshine
Ohh...Umm.. :33

To find the point of intersection, should I make the table of values for each equation ? Or is there another way ?

You don't need to find the intersection point at all: it is a very nasty thing (square roots inside square roots...you don't want to meet such a thing in an obscure alley at night!)!

Tonio

9. Originally Posted by tonio
But you're given what xy is, girl!

Tonio
O_O I am!!

I'm so sorry !! X3

Then it would be -3/3 = -1 which is a choice @_@

Sorry, sorry !

10. That's true. Tonio's way is much easier. You're nearly there.