For what values of k does the line y=x + k pass through the circle defined by x^2 + y^2 = 25 at: 2 points, 1 point, and 0 points
i found the discriminant which is 100 - 4k^2
but im not sure how to find the values for each point
if 100 - 4k^2 > 0 then you get 2 points
if 100 - 4k^2 = 0 then you get 1 point
if 100 - 4k^2 < 0 then you get 0 points
that cant be, because you need to find the values for those number of values
if the discriminant is zero, there is only one root, hence only one point of intersection.
positive, two roots
negative zero roots
Incidentally, you'd better check your discriminant. If you graph y=x+5, you will see that it intersects in two points, not 1.
http://www.wolframalpha.com/input/?i=plot+x%5E2%2By%5E2%3D25%2C+y%3Dx%2B5
No, no it asks you for the values of k that let you get 2pts or 1pt or 0pts.
You have to solve each of those.
E.g.
100 - 4k^2 = 0
100 = 4k^2
25 = k^2
5 = k
So, when k = 5 you get only 1 point
100 - 4k^2 > 0
100 > 4k^2
25 > k^2
5 > k
So, when k > 5 you get 2 points
etc
@ GanonTEK i understand what you mean. i understand what to do now, thank you
You're welcome Jackie. Steve is right about the discriminant being incorrect by the way.
x^2 + y^2 = 25
y = x - k
x^2 + x^2 - 2xk +k^2 = 25
2x^2 - 2xk + k^2 - 25 = 0
x = 2k +- Sqrt(4k^2 - 4(2)(k^2 - 25)))/4
your discriminant is
-4k^2 + 200
To determine how many points of intersection occur between the line y = x + k and the circle x^2 + y^2 = 25, we need to analyze the discriminant of the quadratic equation.
Step 1: Start with the equation of the line: y = x + k
Step 2: Substitute y in the circle equation with x + k: x^2 + (x + k)^2 = 25
Step 3: Simplify the equation: x^2 + (x^2 + 2kx + k^2) = 25
Step 4: Combine like terms: 2x^2 + 2kx + k^2 = 25
Step 5: Rewrite the equation in standard quadratic form: 2x^2 + 2kx + (k^2 - 25) = 0
Now we have a quadratic equation of the form ax^2 + bx + c = 0, where a = 2, b = 2k, and c = k^2 - 25.
The discriminant of this quadratic equation can be calculated using the formula Δ = b^2 - 4ac.
Step 6: Substitute the values of a, b, and c into the discriminant formula: Δ = (2k)^2 - 4(2)(k^2 - 25)
Step 7: Simplify the equation: Δ = 4k^2 - 8(k^2 - 25)
Step 8: Combine like terms: Δ = 4k^2 - 8k^2 + 200
Step 9: Simplify the expression: Δ = -4k^2 + 200
Now, we can analyze the discriminant Δ to determine the number of points of intersection between the line and the circle.
If Δ > 0, there are two distinct points of intersection.
If Δ = 0, there is one point of intersection.
If Δ < 0, there are no points of intersection.
For each case, we substitute the discriminant value into the equation:
Case 1: Δ > 0
-4k^2 + 200 > 0
-4k^2 > -200
k^2 < 50
-√50 < k < √50
Case 2: Δ = 0
-4k^2 + 200 = 0
-4k^2 = -200
k^2 = 50
k = ±√50
Case 3: Δ < 0
-4k^2 + 200 < 0
-4k^2 < -200
k^2 > 50
No solution exists for this case.
Therefore, for two points of intersection, -√50 < k < √50.
For one point of intersection, k = ±√50.
For zero points of intersection, k^2 > 50.