In Exercises 99-102, find an equation of the tangent line to...

In Exercises $99-102,$ find an equation of the tangent line to the circle at the indicated point. Recall from geometry that the tangent line to a circle is perpendicular to the radius of the circle at the point of tangency.
$$
x^{2}+y^{2}=24 \quad(-2 \sqrt{5}, 2)
$$

Key Concepts

Circle

A circle in analytic geometry is defined as the set of all points that are a fixed distance (the radius) away from a fixed point (the center). Its equation in standard form helps identify the center and radius, which are key for understanding its geometric properties.

Tangent Line

A tangent line to a circle is a straight line that touches the circle at exactly one point. At the point of tangency, the tangent line is perpendicular to the radius drawn to that point, a property that is central to determining the tangent line’s equation.

Perpendicularity

Perpendicularity is a critical concept in geometry where two lines meet at a right angle (90 degrees). In the context of a circle, this relationship ensures that the tangent line is orthogonal to the radius at the point of tangency, meaning their slopes are negative reciprocals.

Slope of a Line

The slope of a line measures its steepness and direction. It is calculated as the ratio of the vertical change to the horizontal change between two points. In problems involving tangents to circles, understanding slopes is essential because the slope of the tangent can be derived using the relationship with the slope of the radius.

Equation of a Line

The equation of a line, often expressed in point-slope form or slope-intercept form, represents all the points on the line. Point-slope form is particularly useful when a specific point on the line and the slope are known, such as when finding a tangent line to a circle.

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