There is a problem that I have solved by one method. However, another method does not solve it.
I have assumed the point P to have coordinates (x, y).
By the Distance Formula we get:
. . .Equation 1:
. . .\(\displaystyle \L \sqrt{(x\,-\,c)^2\, +\,y^2}\, +\, \sqrt{(x+c)^2 + y^2}\, =\,2a\)
Then:
. . .\(\displaystyle \L \frac{1}{\sqrt{(x\,-\,c)^2\,+\,y^2}\,+ \,sqrt{(x\,+\,c)^2\,+\,y^2}}\, =\,\frac{1}{2a}\)
Rationaliting the denominator, we get:
sqrt[(x-c)^2 +y^2] - sqrt[(x+c)^2 + y^2]
=>---------------------------------------------------------- = 1/(2a)**********[important1]
-2cx -2cx
sqrt[(x-c)^2 +y^2] - sqrt[(x+c)^2 + y^2] = [-2cx]/a _________________(a2eq)
adding 1eq&2eq we get ; 2sqrt[(x-c)^2 +y^2] = 2(a^2 - cx)/a
now proceeding accordingly we get the result,
However
sqrt[(x+c)^2 +y^2] - sqrt[(x-c)^2 + y^2]
=>---------------------------------------------------------- = 1/(2a)**********[important2]
-2cx -2cx
does not give the result.
Q1) Why could this be, or am I going wrong?
Q2) If I am correct, then what does this dilemma point to?
Regards,
Sujoy
[Nothing more is given; the above is the full statement of the question.]Question: The sum of the distances of a movable point P from the two fixed points
(c, 0) and (-c, 0) is a constant value of 2a. Prove that the the equation to the locus of P is given by:
. . .\(\displaystyle \L \frac{x^2}{a^2}\, +\, \frac{y^2}{b^2}\, =\,1\)
...where a<sup>2</sup> - b<sup>2</sup> = c<sup>2</sup>.
I have assumed the point P to have coordinates (x, y).
By the Distance Formula we get:
. . .Equation 1:
. . .\(\displaystyle \L \sqrt{(x\,-\,c)^2\, +\,y^2}\, +\, \sqrt{(x+c)^2 + y^2}\, =\,2a\)
Then:
. . .\(\displaystyle \L \frac{1}{\sqrt{(x\,-\,c)^2\,+\,y^2}\,+ \,sqrt{(x\,+\,c)^2\,+\,y^2}}\, =\,\frac{1}{2a}\)
Rationaliting the denominator, we get:
sqrt[(x-c)^2 +y^2] - sqrt[(x+c)^2 + y^2]
=>---------------------------------------------------------- = 1/(2a)**********[important1]
-2cx -2cx
sqrt[(x-c)^2 +y^2] - sqrt[(x+c)^2 + y^2] = [-2cx]/a _________________(a2eq)
adding 1eq&2eq we get ; 2sqrt[(x-c)^2 +y^2] = 2(a^2 - cx)/a
now proceeding accordingly we get the result,
However
sqrt[(x+c)^2 +y^2] - sqrt[(x-c)^2 + y^2]
=>---------------------------------------------------------- = 1/(2a)**********[important2]
-2cx -2cx
does not give the result.
Q1) Why could this be, or am I going wrong?
Q2) If I am correct, then what does this dilemma point to?
Regards,
Sujoy