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 conditions under which the line does not intersect the circle.
First, let's rewrite the equation of the line in terms of y: y = x + 7.
Substituting this into the equation of the circle, we get:
x^2 + (x + 7)^2 = 25.
Expanding this equation, we get:
x^2 + (x^2 + 14x + 49) = 25.
Simplifying further, we have:
2x^2 + 14x + 24 = 0.
Now, to find the range of values for k, we need to solve this quadratic equation for x.
Let's consider the discriminant of the quadratic equation, given by the formula:
b^2 - 4ac,
where a, b, and c are the coefficients of the quadratic equation.
For our equation, the coefficients are:
a = 2,
b = 14,
c = 24.
The discriminant then becomes:
(14)^2 - 4(2)(24) = 196 - 192 = 4.
Since the discriminant is positive, the quadratic equation has two distinct real roots.
When the discriminant is positive, the line intersects the circle at two different points.
However, we are interested in finding the range of values for k for which the line DOES NOT intersect the circle.
Therefore, we need to find the values of k for which the quadratic equation does not have real roots, i.e., when the discriminant is negative.
Since the discriminant is positive in this case, we can conclude that there are no values of k for which the circle and the line do not intersect.
Hence, the circle and the line always intersect for any value of k.