What is the focal distance of a point on the hyperbola?
The sum of the focal distance of any point on a hyperbola is constant and equal to the length of the transverse axis of the hyperbola.
Let P (x, y) be any point on the hyperbola \(\frac{x^{2}}{a^{2}}\)  \(\frac{y^{2}}{b^{2}}\) = 1.a
Let MPM' be the perpendicular through P on directrices ZK and Z'K'. Now by definition we get,
SP = e ∙ PM
⇒ SP = e ∙ NK
⇒ SP = e (CN  CK)
⇒ SP = e(x  \(\frac{a}{e}\))
⇒ SP = ex  a ………………..…….. (i)
and
S'P = e ∙ PM'
⇒ S'P = e ∙ (NK')
⇒ S'P = e (CK' + CN)
⇒ S'P = e (\(\frac{a}{e}\) + x)
⇒ S'P = a + ex ………………..…….. (ii)
Therefore, S'P  SP = (a + ex)  (ex  a) = a + ex  ex + a = 2a = transverse axis.
Hence, the sum of the focal distance of a point P (x, y) on the hyperbola \(\frac{x^{2}}{a^{2}}\)  \(\frac{y^{2}}{b^{2}}\) = 1 is constant and equal to the length of the transverse axis (i.e., 2a) of the hyperbola.
Note: This property leads to an alternative definition of hyperbola as follows:
If a point moves on a plane in such a way that the sum of its distances from two fixed points on the plane is always a constant then the locus traced out by the moving point on the plane is called a hyperbola and the two fixed points are the two foci of the hyperbola.
Solved example to find the focal distance of any point on a hyperbola:
Find the focal distance of a point on the hyperbola 9x\(^{2}\)  16y\(^{2}\)  18x + 32y  151 = 0.
Solution:
The given equation of the hyperbola is 9x\(^{2}\)  16y\(^{2}\)  18x + 32y  151 = 0.
From the above equation we get,
9x\(^{2}\)  18x  16y\(^{2}\) + 32y = 151
⇒ 9(x\(^{2}\)  2x)  16(y\(^{2}\)  2y) = 151
⇒ 9(x\(^{2}\)  2x + 1)  16(y\(^{2}\)  2y + 1) = 151 + 9  16
⇒ 9(x  1)\(^{2}\)  16(y  1)\(^{2}\) = 144
⇒ \(\frac{(x  1)^{2}}{16}\)  \(\frac{(y  1)^{2}}{9}\) = 1 ………………….. (i)
Now transfering the origin at (1, 1) without rotating the coordinate axes and denoting the new coordinates with respect to the new axes by x and y, we have
x = X + 1 and y = Y + 1 ………………….. (ii)
Using these relations, equation (i) reduces to
\(\frac{X^{2}}{4^{2}}\)  \(\frac{Y^{2}}{3^{2}}\) = 1 ……………………… (iii)
This is the form of \(\frac{X^{2}}{a^{2}}\)  \(\frac{Y^{2}}{b^{2}}\) = 1 (a\(^{2}\) > b\(^{2}\) ) where a = 4 and b = 3
Now, we get that a > b.
Hence, the equation\(\frac{X^{2}}{4^{2}}\)  \(\frac{Y^{2}}{3^{2}}\) = 1 represents an hyperbola
whose transverse axes along X and conjugate axes along Y axes.
Therefore, the focal distance of a point on the hyperbola 9x\(^{2}\)  16y\(^{2}\)  18x + 32y  151 = 0 is transverse axis = 2a = 2 ∙ 4 = 8 units.
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