Final answer:
There are 3 integral values of k (-2, -1, and 0) for which no tangent can be drawn from the point (k, k+2) to the circle x² + y² = 4, as the point must be inside or on the circle.
Step-by-step explanation:
To determine the number of integral values of k for which no tangent can be drawn from the point (k, k+2) to the circle x² + y² = 4, we need to consider the distance from the point to the center of the circle, which is at (0,0). A tangent can be drawn to a circle from a point outside it if the distance from the point to the center of the circle is greater than the radius of the circle.
The equation of the circle given is x² + y² = 4, which implies a radius of 2 units. For the point (k, k+2), we calculate the distance to the origin using the distance formula:
d = √((k-0)² + (k+2-0)²) = √(k² + k² + 4k + 4) = √(2k² + 4k + 4)
We now require that this distance d be less than or equal to 2 (the radius of the circle), as no tangent can be drawn if the point is inside or on the circle:
√(2k² + 4k + 4) ≤ 2
By squaring both sides of the inequality and simplifying:
2k² + 4k + 4 ≤ 4
2k² + 4k ≤ 0
Dividing everything by 2:
k² + 2k ≤ 0
This can be factored as:
k(k + 2) ≤ 0
The inequality is satisfied for values of k between -2 and 0, including -2 and 0. Thus, the integral values of k that satisfy this inequality are -2, -1, and 0. So, the answer is that there are 3 integral values of k for which no tangent can be drawn from the point to the circle.