First Principle Hyperbolas
Date: 01/05/99 at 08:58:55 From: Janis Subject: Hyperbolas Doctor Math, I'd like to know the answer to this question. How do you solve for the standard equation of a hyperbola given its properties? I know the standard equation already because we have discussed it at school. I just hope that your answer will come in a step-by-step process. Thanks a lot in advance, Janis
Date: 01/05/99 at 10:27:56 From: Doctor Rob Subject: Re: Hyperbolas Thanks for writing to Ask Dr. Math! I assume that by "its properties" you mean that a hyperbola is the locus of points such that the absolute difference between the distances from the two foci is a constant. In other words, if you take any point on the hyperbola, measure how far it is from each of the foci, and take the absolute difference of the two measurements, you will get the same constant. There are other definitions, and they lead to different derivations. Let the foci F and G have coordinates (-c,0) and (c,0), where c > 0 (if c = 0, the foci coincide, and the locus is either the entire xy-plane or empty). Let the general point P on the locus have coordinates (x,y). Let the constant be 2*a, where a >= 0 (you'll see that the 2 makes the equation come out nicer in the end). Then by considering the triangle PFG and using the Triangle Inequality, PF + FG >= PG, and PG + FG >= PF FG >= PG - PF, and FG >= PF - PG 2*c = FG >= |PF-PG| = 2*a c >= a Now use the distance formula to compute the distances PF and PG. Then, by the above locus definition |sqrt([x+c]^2+y^2)-sqrt([x-c]^2+y^2)| = 2*a Square both sides, use the fact that |Q|^2 = Q^2 for any real number Q, and expand [x+c]^2+y^2-2*sqrt([x+c]^2+y^2)*sqrt([x-c]^2+y^2)+[x-c]^2+y^2 = 4*a^2 Isolate the term with the square roots on the righthand side, gather like terms, and divide everything by 2: x^2 + c^2 + y^2 - 2*a^2 = sqrt([x+c]^2+y^2)*sqrt([x-c]^2+y^2) Now square again, and all the square roots will go away: (x^2+c^2+y^2-2*a^2)^2 = ([x+c]^2+y^2)*([x-c]^2+y^2) Now expand everything in sight, and move it all to the lefthand side, gathering like terms. Divide by 4, and you should have (c^2-a^2)*x^2 - a^2*y^2 - a^2*(c^2-a^2) = 0 Now recall that c >= a, so c^2 - a^2 >= 0. Put b = sqrt(c^2-a^2), so b >= 0, substitute c^2 - a^2 = b^2, and add a^2*b^2 to both sides to get b^2*x^2 - a^2*y^2 = a^2*b^2 At this point, everything we have done is fine. Now we want to divide by a^2*b^2, to get the locus equation x^2/a^2 - y^2/b^2 = 1 which is the standard form. This works if and only if both a and b are nonzero, if and only if c > a > 0. This is the answer you seek. If a = 0, then b = c > 0, and we get the locus equation x^2 = 0, which is two copies of the line x = 0; that is, two coincident lines. If a > 0 and b = 0 (so c = a), then we get the locus equation y^2 = 0, which is two copies of the line y = 0, again two coincident lines. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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