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.
$$
x^{2}+y^{2}=12
$$

Key Concepts

Point-Slope Form

The point-slope form of a line, expressed as y - y? = m(x - x?), is a method used to write the equation of a line when a point on the line and its slope are known. This form is particularly useful in problems involving tangents, as once the slope of the tangent is determined and the point of tangency is given, the equation of the tangent line can be easily formulated.

Perpendicularity of the Radius and Tangent

A fundamental geometric property of circles is that the radius drawn to the point of tangency is perpendicular to the tangent line at that point. This principle is vital in solving problems involving tangents, because it allows one to relate the slope of the radius to the slope of the tangent through the concept that the slopes of perpendicular lines are negative reciprocals of each other.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations that are not solved for one variable in terms of another. For curves defined by equations like those of circles, implicit differentiation enables the calculation of the derivative (slope) at any point on the curve, which is crucial for finding the slope of the tangent line.

Circle

A circle is the set of all points in a plane that are at a constant distance (the radius) from a fixed point (the center). Understanding the equation of a circle, typically expressed in the form x² + y² = r², is essential because it lays the groundwork for analyzing its geometric properties, including tangency and symmetry.

Tangent Line

A tangent line to a curve is a straight line that touches the curve at exactly one point and has the same instantaneous direction as the curve at that point. In the context of circles, the tangent line is key because it meets the circle precisely once at the point of tangency and does not intersect the circle at nearby points.

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