If a line y = 2x + k touches circle x² + y² = 9, find k.

Steps to Solve

1. Substitute y from the line equation into the circle equation

We have the line equation $y = 2x + k$. We'll substitute this into the circle equation $x^2 + y^2 = 9$.

Substituting gives us:

$$x^2 + (2x + k)^2 = 9$$

2. Expand the equation

Now, we need to expand the equation obtained from the substitution:

$$(2x + k)^2 = 4x^2 + 4kx + k^2$$

So the equation becomes:

$$x^2 + 4x^2 + 4kx + k^2 = 9$$

Combine the like terms:

$$5x^2 + 4kx + (k^2 - 9) = 0$$

3. Set the discriminant to zero for tangency

For the line to be tangent to the circle, the quadratic equation must have exactly one solution. This occurs when the discriminant ($D$) is equal to zero.

The discriminant for our quadratic equation $ax^2 + bx + c = 0$ is given by:

$$D = b^2 - 4ac$$

Here, $a = 5$, $b = 4k$, and $c = k^2 - 9$. We set the discriminant to zero:

$$(4k)^2 - 4(5)(k^2 - 9) = 0$$

4. Solve the discriminant equation

Now we need to solve:

$$16k^2 - 20(k^2 - 9) = 0$$

Distributing gives us:

$$16k^2 - 20k^2 + 180 = 0$$

Combine like terms:

$$-4k^2 + 180 = 0$$

Now, solving for $k^2$:

$$4k^2 = 180$$

$$k^2 = 45$$

Take the square root:

$$k = \pm \sqrt{45} = \pm 3\sqrt{5}$$