The curve `xy = c(c > 0)` and the circle `x^2 +y^2=1` touch at two points, then distance between the points of contact is

Options

A. (a) `1`
B. (b) `2`
C. (c) `2sqrt2`
D. (d) none of these

Explanation

To find the distance between the points of contact of the curve \(xy = c\) (where \(c > 0\)) and the circle \(x^2 + y^2 = 1\), we will follow these steps:

### Step 1: Substitute \(y\) from the hyperbola into the circle's equation.
Given the hyperbola \(xy = c\), we can express \(y\) in terms of \(x\):
\[ y = \frac{c}{x} \]
Now substitute this into the equation of the circle:
\[ x^2 + \left(\frac{c}{x}\right)^2 = 1 \]

### Step 2: Simplify the equation.
Substituting \(y\) gives:
\[ x^2 + \frac{c^2}{x^2} = 1 \]
Multiply through by \(x^2\) to eliminate the fraction:
\[ x^4 - x^2 + c^2 = 0 \]

### Step 3: Let \(u = x^2\) and rewrite the equation.
Let \(u = x^2\), then the equation becomes:
\[ u^2 - u + c^2 = 0 \]

### Step 4: Find the discriminant.
For the two curves to touch at two points, the discriminant of this quadratic must be zero:
\[ D = b^2 - 4ac = (-1)^2 - 4(1)(c^2) = 1 - 4c^2 = 0 \]
Setting the discriminant to zero gives:
\[ 1 - 4c^2 = 0 \implies 4c^2 = 1 \implies c^2 = \frac{1}{4} \implies c = \frac{1}{2} \]

### Step 5: Solve for \(x^2\).
Substituting \(c = \frac{1}{2}\) back into the quadratic:
\[ u^2 - u + \frac{1}{4} = 0 \]
The roots can be calculated as:
\[ u = \frac{1 \pm 0}{2} = \frac{1}{2} \]
Thus, \(x^2 = \frac{1}{2}\) implies:
\[ x = \pm \frac{1}{\sqrt{2}} \]

### Step 6: Find corresponding \(y\) values.
Using \(xy = \frac{1}{2}\):
\[ y = \frac{c}{x} = \frac{1/2}{\pm \frac{1}{\sqrt{2}}} = \pm \frac{1}{\sqrt{2}} \]
Thus, the points of contact are:
\[ \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \quad \text{and} \quad \left(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right) \]

### Step 7: Calculate the distance between the points.
The distance \(d\) between the points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the points:
\[ d = \sqrt{\left(-\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right)^2 + \left(-\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right)^2} \]
This simplifies to:
\[ d = \sqrt{\left(-\frac{2}{\sqrt{2}}\right)^2 + \left(-\frac{2}{\sqrt{2}}\right)^2} = \sqrt{2 \cdot \left(-\frac{2}{\sqrt{2}}\right)^2} = \sqrt{2 \cdot \frac{4}{2}} = \sqrt{4} = 2 \]

### Final Answer:
The distance between the points of contact is \(2\).

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