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An ellipse is a geometrical shape that appears in two dimensions as an elongated circle. It's defined mathematically by an equation of the form:

• Standard form of ellipse centered at the origin: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
• If placed in the form: \[ Ax^2 + By^2 = C \], then the ellipse is oriented along the x and y axes.

For the ellipse in our exercise, the equation given is \(x^2 + 4y^2 = 8\). This tells us it is centered at the origin \((0,0)\), and has different values for eccentricities along the x and y-axis, making it an elongated circle.
The coefficients in the equation signify the shape of the ellipse. Here, the coefficient of 4 in front of \(y^2\) means the ellipse stretches more along the y-axis.
An important property of ellipses is the concept of tangents – lines that just touch the ellipse at one point without crossing it. Such lines are crucial for understanding how different shapes interact geometrically.

The discriminant is a key concept in analyzing quadratic equations. It helps us understand the nature of the roots without solving the equation directly.
For any quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant is denoted as:

• \( \Delta = b^2 - 4ac \)

Depending on the value of the discriminant, the nature of the roots varies:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root, indicating the presence of a tangent.
• If \( \Delta < 0 \), the equation has no real roots, pointing to no real intersection with the x-axis.

In this exercise, finding the discriminant for the quadratic \(2x^2 - 2kx + k^2 - 8 = 0\) and setting it to zero ensures that the line is tangent to the ellipse. This results in finding those specific values of \(k\) that make the line just touch the ellipse at one point.

Quadratic equations form an essential part of algebra and geometry. They appear in various scientific fields and describe phenomena involving parabolic paths.
The general form of a quadratic equation is:

• \( ax^2 + bx + c = 0 \)

These equations feature three coefficients \(a\), \(b\), and \(c\) which dictate the shape and position of the parabola. Here, the term quadratic implies that the variable \(x\) is raised to the power of two.
Solving quadratic equations can be done through:

• Factoring, if easily possible
• Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
• Completing the square, another algebraic approach

In the context of our exercise, the quadratic is derived from substituting the equation of the line into the ellipse. Simplifying this substitution yields \(2x^2 - 2kx + k^2 - 8 = 0\), a quadratic in terms of \(x\), which helps in determining the values of \(k\) for which the line acts as a tangent to the ellipse.