Solve the given problems involving tangent and normal lines....

solve the given problems involving tangent and normal lines.
Find the equations of the tangent and normal lines to the ellipse with focus at $(4,0),$ vertex at $(5,0),$ and center at the origin, where $x=2 .$ Use a calculator to view the curve and lines.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent variable is not isolated on one side of the equation. This method allows for the computation of derivatives for curves defined by implicit relationships, such as the equation of an ellipse, thereby facilitating the determination of slopes of tangent and normal lines.

Normal Line

A normal line to a curve at a given point is the line that is perpendicular to the tangent line at that point. Its slope is the negative reciprocal of the slope of the tangent line. Normal lines are important in many applications, including physics and optimization.

Ellipse

An ellipse is a type of conic section characterized by two fixed points known as foci. The sum of the distances from any point on the ellipse to each focus is constant. The ellipse has a center, vertices, and axes (major and minor), and its standard equation can be derived from these geometric properties. The positions of the foci and vertices help in determining crucial parameters necessary for defining its precise shape.

Ellipse Equation Parameters

The equation of an ellipse with a center at the origin is generally expressed as x²/a² + y²/b² = 1, where 'a' represents the semi-major axis and 'b' the semi-minor axis. The distance 'c' from the center to a focus satisfies the relationship c² = a² - b². These parameters are interrelated, and knowing certain features of an ellipse, such as a vertex or a focus, provides the information needed to calculate the remaining parameters.

Tangent Line

A tangent line to a curve is a straight line that touches the curve at one point without crossing it in the vicinity of that point. Its slope is equal to the derivative of the curve evaluated at the point of tangency. The concept of a tangent is fundamental to understanding local linear approximations of curves.

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