If the tangent to the ellipse x^2+4 y^2=16 at the point with...

If the tangent to the ellipse $x^{2}+4 y^{2}=16$ at the point with eccentric angle $\phi$ is a normal to the curve $x^{2}+y^{2}-8 x-4 y$ $=0$, then $\phi$ equals
(a) $\frac{\pi}{3}$
(b) $\frac{-\pi}{3}$
(c) $\frac{\pi}{6}$
(d) $\frac{\pi}{2}$

Key Concepts

Ellipse Parameterization and Eccentric Angle

An ellipse in standard form can be parameterized using the eccentric angle, where the coordinates of a point on the ellipse are expressed as (a cos ?, b sin ?) with a and b being the semi-axes lengths. This parameterization is useful because it provides a direct connection between the geometric properties of the ellipse and the trigonometric functions, simplifying the analysis of properties such as tangents and normals at a particular point on the ellipse.

Tangent to an Ellipse

The tangent line to an ellipse at a given point is a line that locally touches the ellipse without intersecting it, and its equation can be derived using the derivative of the ellipse’s parametric form or by employing the point form. The slope of this tangent line is related to the derivatives of the parameterized coordinates, and understanding its formulation is crucial when establishing relationships with other curves.

Normal to a Circle

A normal to a circle at a given point is the line that passes through the point of tangency and the center of the circle. Since every radius drawn to the circle is perpendicular to the tangent at the point of contact, the condition for a given line to be normal to a circle requires that it passes through the circle’s center. This property is used to connect the conditions of a tangent to one curve and a normal to another.

Use of Slope and Perpendicularity Conditions

Analyzing the relationships between tangents and normals involves calculating and comparing slopes. For instance, if a line is known to be tangent to one curve and normal to another, the slope conditions (either through differentiation or geometric relationships) must satisfy both sets of requirements. This technique is central when solving problems that involve multiple conic sections and their interrelated line properties.

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