find the range of values for k for which the circle x^2+y^2=25 and the line y=x+7 do not intersect
To find the range of values for k for which the circle and the line do not intersect, we need to determine the positions of the line with respect to the circle.
The equation of the circle is x^2 + y^2 = 25.
The equation of the line is y = x + 7.
To analyze the positions of the line and the circle, we need to substitute the equation of the line into the equation of the circle and solve for x.
Substituting y = x + 7 in the equation of the circle, we get:
x^2 + (x + 7)^2 = 25
Expanding and simplifying the equation, we have:
x^2 + x^2 + 14x + 49 = 25
2x^2 + 14x + 24 = 0
Dividing all terms by 2, we have:
x^2 + 7x + 12 = 0
Now we need to find the values of x that satisfy this equation, which will give us an idea of the range of values for k where the line and the circle do not intersect.
We can either factor this quadratic equation or apply the quadratic formula to find the solutions. Let's use factoring:
(x + 3)(x + 4) = 0
Setting each factor to zero, we get:
x + 3 = 0 --> x = -3
x + 4 = 0 --> x = -4
So, the line intersects the circle at x = -3 and x = -4.
To determine the range of values for k, we need to consider the y-coordinate of the line for these x-values.
For x = -3, y = -3 + 7 = 4.
For x = -4, y = -4 + 7 = 3.
Therefore, when the line intersects the circle, the coordinates of the intersection points are (-3, 4) and (-4, 3).
So, the range of values for k for which the circle and the line do not intersect is:
k < 3 or k > 4