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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 x2a2 - y2b2 = 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 - ae)
⇒ SP = ex - a ………………..…….. (i)
and
S'P = e ∙ PM'
⇒ S'P = e ∙ (NK')
⇒ S'P = e (CK' + CN)
⇒ S'P = e (ae + 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 x2a2 - y2b2 = 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 9x2 - 16y2 - 18x + 32y - 151 = 0.
Solution:
The given equation of the hyperbola is 9x2 - 16y2 - 18x + 32y - 151 = 0.
From the above equation we get,
9x2 - 18x - 16y2 + 32y = 151
⇒ 9(x2 - 2x) - 16(y2 - 2y) = 151
⇒ 9(x2 - 2x + 1) - 16(y2 - 2y + 1) = 151 + 9 - 16
⇒ 9(x - 1)2 - 16(y - 1)2 = 144
⇒ (x−1)216 - (y−1)29 = 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
X242 - Y232 = 1 ……………………… (iii)
This is the form of X2a2 - Y2b2 = 1 (a2 > b2 ) where a = 4 and b = 3
Now, we get that a > b.
Hence, the equationX242 - Y232 = 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 9x2 - 16y2 - 18x + 32y - 151 = 0 is transverse axis = 2a = 2 ∙ 4 = 8 units.
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