To find the exact points at which the circle $x^2 + y^2 = 42$ intersects the straight line $y = 2x$, we can follow these steps:

Step 1: Substitute the equation of the line into the equation of the circle.

The equation of the circle is:

$x^2 + y^2 = 42$

Substitute $y = 2x$ into the equation of the circle:

$x^2 + (2x)^2 = 42$

Step 2: Simplify and solve the quadratic equation.

$x^2 + 4x^2 = 42$

$5x^2 = 42$

$x^2 = \frac{42}{5}$

$x = \pm \sqrt{\frac{42}{5}} = \pm \frac{\sqrt{210}}{5}$

Step 3: Find the corresponding $y$ values.

Since $y = 2x$, substitute the $x$ values back into the equation for $y$:

For $x = \frac{\sqrt{210}}{5}$:

$y = 2 \times \frac{\sqrt{210}}{5} = \frac{2\sqrt{210}}{5}$

For $x = -\frac{\sqrt{210}}{5}$:

$y = 2 \times \left(-\frac{\sqrt{210}}{5}\right) = -\frac{2\sqrt{210}}{5}$

Step 4: Write the intersection points.

The points of intersection are:

$\left(\frac{\sqrt{210}}{5}, \frac{2\sqrt{210}}{5}\right) \quad \text{and} \quad \left(-\frac{\sqrt{210}}{5}, -\frac{2\sqrt{210}}{5}\right)$

These are the exact points at which the circle intersects the line.

Would you like further details on this solution, or do you have any questions?

Here are some related questions:

1. How do you determine the radius of a circle given its equation?
2. What are the general forms of the equations of a line and a circle?
3. How can you find the intersection points of two circles?
4. What does it mean geometrically when a line intersects a circle at exactly one point?
5. How can we determine if a line is tangent to a circle?

Tip: When solving intersection problems, always start by substituting the equation of one curve into the other, which reduces the problem to solving a single equation.