Find the values of k for which the line y=kx-2 meets the curve y^2=4x-x^2

To find the values of \( k \) for which the line \( y = kx - 2 \) intersects the curve \( y^2 = 4x - x^2 \), we need to solve the system of equations by substituting the equation of the line into the equation of the curve.

Given:
\[
y = kx - 2
\]
\[
y^2 = 4x - x^2
\]

### Step 1: Substitute \( y = kx - 2 \) into \( y^2 = 4x - x^2 \).

Substitute \( y = kx - 2 \) into \( y^2 = 4x - x^2 \):
\[
(kx - 2)^2 = 4x - x^2
\]

### Step 2: Expand and simplify the equation.

Expanding the left side:
\[
(kx - 2)^2 = k^2x^2 - 4kx + 4
\]

Now, equate this to the right side:
\[
k^2x^2 - 4kx + 4 = 4x - x^2
\]

### Step 3: Move all terms to one side to form a quadratic equation in \( x \).

Rearrange the equation:
\[
k^2x^2 + x^2 - 4kx - 4x + 4 = 0
\]

Combine like terms:
\[
(k^2 + 1)x^2 - (4k + 4)x + 4 = 0
\]

### Step 4: Solve the quadratic equation for discriminant.

For the line to intersect the curve, the quadratic equation must have real solutions. The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[
\Delta = b^2 - 4ac
\]

Here, \( a = k^2 + 1 \), \( b = -(4k + 4) \), and \( c = 4 \).

Substitute these into the discriminant formula:
\[
\Delta = (-(4k + 4

Explore how to find values of k for which the line y=kx-2 intersects the curve y^2=4x-x^2. Learn the quadratic equation and discriminant formula application, essential for understanding algebraic intersections. Suitable for advanced high school students and beyond.

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