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Chapter 15: Problem 1464

The equation of the chord joining two points \(\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) and \(\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)\) on the rectangular hyperbola \(\mathrm{xy}=\mathrm{c}^{2}\) is (a) \(\left[x /\left(y_{1}+y_{2}\right)\right]-\left[y /\left(x_{1}+x_{2}\right)\right]=1\) (b) \(\left[x /\left(x_{1}-x_{2}\right)\right]+\left[y /\left(y_{1}-y_{2}\right)\right]=1\) (c) \(\left[x /\left(y_{1}-y_{2}\right)\right]-\left[y /\left(x_{1}-x_{2}\right)\right]=1\) (d) \(\left[\mathrm{x} /\left(\mathrm{x}_{1}+\mathrm{x}_{2}\right)\right]+\left[\mathrm{y} /\left(\mathrm{y}_{1}+\mathrm{y}_{2}\right)\right]=1\)

Short Answer

Expert verified
The short answer is : The equation of the chord joining two points \((x_1, y_1)\) and \((x_2, y_2)\) on the rectangular hyperbola \(xy = c^2\) is \[\frac{x}{(y_1 - y_2)} - \frac{y}{(x_1 - x_2)} = 1\] which matches option (c).

Step by step solution

01

Identify the points on the hyperbola

First, the given rectangular hyperbola is \(xy = c^2\). The two points on the hyperbola are \((x_1, y_1)\) and \((x_2, y_2)\). Note that these points satisfy the equation of the hyperbola. That is, \(x_1 y_1 = c^2\) and \(x_2 y_2 = c^2\).
02

Find the slope of the chord

We need to find the slope of the line connecting the two points. The slope is given by \[\frac{y_2 - y_1}{x_2 - x_1}\]
03

Use the slope-point form to find the equation of the chord

Using the slope-point formula, which is \[y - y_1 = m(x - x_1)\], where m is the slope of the line and \((x_1, y_1)\) is one of the points on the line, we have: \[y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\]
04

Simplify the chord's equation

Now, we will simplify the equation to derive which of the options is the correct one. Rearranging and simplifying, we have: \[x(y_2 - y_1) - y(x_2 - x_1) = x_1 y_1 - x_2 y_1\] Now, recalling that \(x_1y_1 = c^2\) and \(x_2y_2 = c^2\), we can substitute these expressions into the equation: \[x(y_2 - y_1) - y(x_2 - x_1) = (x_2y_2 - x_1y_1)\] \[x\left(\frac{y_2 - y_1}{x_2 - x_1}\right) - y\left(\frac{x_2 - x_1}{y_2 - y_1}\right) = 1\] This equation can be rewritten as: \[\frac{x}{(y_1 - y_2)} - \frac{y}{(x_1 - x_2)} = 1\] Comparing this equation with the given options, we can see that it matches option (c). Thus, the correct answer is (c).

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Most popular questions from this chapter

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