Number of integral value(s) of k for which no tangent can be drawn from the point `(k, k+2)` to the circle `x^(2)+y^(2)=4` is :

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to find the number of integral values of \( k \) for which no tangent can be drawn from the point \( (k, k+2) \) to the circle defined by the equation \( x^2 + y^2 = 4 \).

### Step-by-Step Solution:

1. **Understand the Circle**: The equation \( x^2 + y^2 = 4 \) represents a circle centered at the origin \( (0, 0) \) with a radius of \( 2 \).

2. **Identify the Point**: The point from which we want to draw tangents is \( (k, k+2) \).

3. **Condition for No Tangent**: A tangent cannot be drawn from a point to a circle if the point lies inside the circle. This means we need to check when the distance from the center of the circle to the point \( (k, k+2) \) is less than the radius of the circle.

4. **Calculate the Distance**: The distance \( d \) from the center \( (0, 0) \) to the point \( (k, k+2) \) is given by:
\[ d = \sqrt{k^2 + (k + 2)^2} \]
Simplifying this:
\[ d = \sqrt{k^2 + (k^2 + 4k + 4)} = \sqrt{2k^2 + 4k + 4} \]

5. **Set Up the Inequality**: For the point to be inside the circle, we need:
\[ d < 2 \]
Squaring both sides (since both sides are positive):
\[ 2k^2 + 4k + 4 < 4 \]
Simplifying this gives:
\[ 2k^2 + 4k < 0 \]

6. **Factor the Inequality**: Dividing the entire inequality by 2 (which does not change the inequality since 2 is positive):
\[ k^2 + 2k < 0 \]
Factoring gives:
\[ k(k + 2) < 0 \]

7. **Determine the Intervals**: The roots of the equation \( k(k + 2) = 0 \) are \( k = 0 \) and \( k = -2 \). The sign of the product \( k(k + 2) \) changes at these points. Testing intervals:
- For \( k < -2 \): both factors are negative, so the product is positive.
- For \( -2 < k < 0 \): the first factor is negative and the second is positive, so the product is negative.
- For \( k > 0 \): both factors are positive, so the product is positive.

Thus, the solution to the inequality \( k(k + 2) < 0 \) is:
\[ -2 < k < 0 \]

8. **Find Integral Values**: The integral values of \( k \) that satisfy \( -2 < k < 0 \) are:
- The only integer in this range is \( -1 \).

### Conclusion:
The number of integral values of \( k \) for which no tangent can be drawn from the point \( (k, k+2) \) to the circle \( x^2 + y^2 = 4 \) is **1**.