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Chapter 15: Problem 1452

If \(\mathrm{x}=9\) is the chord of the hyperbola \(\mathrm{x}^{2}-\mathrm{y}^{2}=9\) then the equation of the corresponding pair of tangents at the end points of the chord is ......... (a) \(9 x^{2}-8 y^{2}+18 x-9=0\) (b) \(9 x^{2}-8 y^{2}-18 x+9=0\) (c) \(9 x^{2}-8 y^{2}-18 x-9=0\) (d) \(9 \mathrm{x}^{2}-8 \mathrm{y}^{2}+18 \mathrm{x}+9=0\)

Short Answer

Expert verified
The short answer is: (a) \(9x^2 - 8y^2 + 18x - 9 = 0\)

Step by step solution

01

Write down the given hyperbola equation

The given equation of the hyperbola is: \(x^2 - y^2 = 9\)
02

Understand the chord condition

It is given that x=9 is the chord of the hyperbola. This means that there exists two points on the hyperbola where x-coordinate of both the points is 9.
03

Calculate the points on the hyperbola

To find these points, we will substitute x=9 in the equation of the hyperbola: \(9^2 - y^2 = 9\) \(81 - y^2= 9\) \(y^2 = 72\) \(y = \pm \sqrt{72}\) So, the end points of the chord are (9, \(\sqrt{72}\)) and (9, \(-\sqrt{72}\)).
04

Calculate the equation of the tangents at the end points

The general equation of a tangent to the hyperbola \(x^2 - y^2 = a^2\) is given by: \(xx_1 - yy_1 = a^2\) Where (x1, y1) are the coordinates of a point on the hyperbola. Now, we will find the equations of the tangents at the end points of the chord. For the point (9, \(\sqrt{72}\)): \(9x-\sqrt{72}y=a^2\) For the point (9, \(-\sqrt{72}\)): \(9x+\sqrt{72}y=a^2\) To find the combined equation of these two tangents, we will eliminate a^2 by subtracting the second equation from the first equation: \((9x-\sqrt{72}y) -(9x+\sqrt{72}y) = 0\)
05

Check for the matching answer

Now we simplify the equation: \(-2\sqrt{72}y = 0\) \(9x^2 - 8y^2 - 18x + 9 = 0\) Comparing this with the given options, the correct answer is: (a) \(9x^2 - 8y^2 + 18x - 9 = 0\)

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