Thread: Proof involving hyberbola

Yes you ar on the right track but to get from line 1 to line 2 you were multiplying throughout by a^2b^2. So instead of =1 you should have =a^2b^2
ax^2+bx+c=0 has repeated roots if b^2-4ac=0 so apply this to your corrected equation.

You are multiplying both sides of the equation by a^2b^2 so in your second line you should have =a^2b^2 instead of =1
The quadratic ax^2+bx+c=0 has repeated roots if b^2-4ac=0 so apply this to your quadratic.

OK, here goes:

a = (b² - m²a²)
b = 2mca²
c = (a²b² - c²)

For repeated roots, b² = 4ac

4m²c²a² = 4.(b² - m²a²)(a²b² - c²)

If I try to simplify and solve this equation, I don't get the result b² - 4ac = 0

If I simply ignore the constants a,b & c, I get the equation x²+2x+1 = 0, for which I get the roots (x+1) & (x+1)

Is this correct?

yes right track but your second line should be=a^2b^2 (You were multiplying both sides by a^2b^2)
Now the general quadratic ax^2+bx+c=0 has repeated roots if b^2-4ac=0

So,



a = (b² - m²a²)
b = -2mca²
c = (- a²c² - a²b²)

b² = 4ac

4m²c²a⁴ = 4.(b² - m²a²).(-a²c² - a²b²)

4m²c²a⁴ = 4[ -a²b²c² + m²a⁴c² - a²b⁴ + m²a⁴b²]

The two sides are not equal.

We get 4m²c²a⁴ on both sides, but what to do with the other terms?

That 1 in your second line should be a^2b^2 (You were in the process of multiplying both sides by a^2b^2
So line should read (b^2-a^2m^2)x^2-2a^2mcx-a^2(b^2+c^2)=0
Now ax^2+bx+c=0 has repeated roots if b^2-4ac=0
So in our equation want (2a^2mc)^2+4(b^2-a^2m^2)a^2(b^2+c^2)=0

That 1 in your second line should be (You were in the process of multiplying both sides by
So line should read
Now has repeated roots if
So in our equation want

I have carried out the correction of multiplying RHS also by and started with the corrected equation,



which can be rewritten as



I have carried out the multiplication as you have said. Incidentally, the equation should be



and not



because the term "C" is negative.

which I have correctly done. How does all this relate to the original equation ?

Edit: Whether the term is negative or positive, it does not matter since there are uncancelled terms on the RHS.

This is correct. Now multiply out brackets. There will be an a^2 which can be cancelled out of every term (since right hand side=0)
You should also see among the terms a^2m^2c^2-a^2m^2c^2, so these go.
Now you should see that there is b^2 which can be cancelled.
Should now have b^2+c^2-a^2m^2=0
Now replace c and m by what we let them stand for at the beginning.

OK, I got it, thanks again for your patience.

Well done! Incidentally in your last post you said that the 'C' is negative which is correct but the formula for repeated roots is B^2-4AC so we had two minus signs which is why I finished with a plus sign.