Question

При каких положительный значение К прямая у=kx-4 имеет с параболой y=x^2-2x ровно одну общую точку? Найдите координаты этой точки.

Answer 1

Kx-4=x^2-2x
x^2-(2+k)x+4=0
D=(2+k)^2-16
4+4k+k^2-16=0
k^2+4k-12=0
D=16+48=64
k1= (-4+8)/2=2
k2=(-4-8)/2= -6(не подходит по условию, k>0)
2x-4=x^2-2x
x^2-4x+4=0
(x-2)^2=0
x=2

Answer 2

Finding the Intersection Point

To find the coordinates of the intersection point between the line and the parabola, we can set their equations equal to each other and solve for the values of x and y.

The given line equation is y = kx - 4 and the parabola equation is y = x^2 - 2x.

By setting these two equations equal to each other, we get: kx - 4 = x^2 - 2x

Now, we can solve for the values of x and y to find the intersection point.

Solving for the Intersection Point

To find the x-coordinate of the intersection point, we can rearrange the equation and solve for x: x^2 - 2x - kx + 4 = 0

This is a quadratic equation in terms of x, and we can solve for x using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)

Where a = 1, b = -2 - k, and c = 4.

After finding the x-coordinate, we can substitute it back into either the line or the parabola equation to find the corresponding y-coordinate.

Conclusion

By solving the quadratic equation and substituting the x-coordinate back into one of the original equations, we can find the coordinates of the intersection point between the line and the parabola.