From the point (2, 2) tangent are drawn to the hyperbola `(x^2)/(16)-(y^2)/9=1.` Then the point of contact lies in the (a) first quadrant (b) second quadrant (c) third quadrant (d) fourth quadrant

Options

A. first quadrant
B. second quadrant
C. third quadrant
D. forth quadrant

Explanation

To solve the problem, we need to determine the quadrant in which the points of contact of the tangents drawn from the point (2, 2) to the hyperbola \(\frac{x^2}{16} - \frac{y^2}{9} = 1\) lie.

### Step 1: Identify the hyperbola parameters
The given hyperbola is \(\frac{x^2}{16} - \frac{y^2}{9} = 1\).

From this equation, we can identify:
- \(a^2 = 16\) → \(a = 4\)
- \(b^2 = 9\) → \(b = 3\)

### Step 2: Write the equations of the asymptotes
The equations of the asymptotes for the hyperbola are given by:
\[ y = \pm \frac{b}{a} x \]

Substituting the values of \(a\) and \(b\):
\[ y = \pm \frac{3}{4} x \]

This gives us two asymptotes:
1. \(y = \frac{3}{4} x\)
2. \(y = -\frac{3}{4} x\)

### Step 3: Determine the slopes of the tangents
The slopes of the tangents drawn from a point \((x_1, y_1)\) to the hyperbola can be found using the formula:
\[ m = \frac{y_1 \pm b\sqrt{(x_1^2/a^2) - 1}}{x_1 - a\sqrt{(y_1^2/b^2) - 1}} \]

Substituting \((x_1, y_1) = (2, 2)\), \(a = 4\), and \(b = 3\):
\[ m = \frac{2 \pm 3\sqrt{(2^2/16) - 1}}{2 - 4\sqrt{(2^2/9) - 1}} \]

### Step 4: Calculate the discriminant
To find the points of contact, we need to ensure that the discriminant of the quadratic formed by the tangent equations is non-negative. This will help us determine the nature of the tangents.

### Step 5: Analyze the position of the point (2, 2) relative to the asymptotes
The point (2, 2) lies above the line \(y = \frac{3}{4} x\) and below the line \(y = -\frac{3}{4} x\). Therefore, the tangents from this point will intersect the hyperbola in the third and fourth quadrants.

### Conclusion
Since the tangents from the point (2, 2) to the hyperbola intersect the hyperbola in the third and fourth quadrants, the points of contact will lie in:

**Answer:** (c) third quadrant and (d) fourth quadrant.