If x cosα+y sinα=p touches x^2+a^2 y^2=a^2, then (a) p^2=a^2 sin^2 α+cos^2 α(b) p^2=a^2 cos^2 α+

step 1 In this problem, note that y from the first equation can be written as p by sine alpha minus x,

If x cosα+y sinα=p touches x^2+a^2 y^2=a^2, then (a) p^2=a^2...

If $x \cos \alpha+y \sin \alpha=p$ touches $x^{2}+a^{2} y^{2}=a^{2}$, then (a) $p^{2}=a^{2} \sin ^{2} \alpha+\cos ^{2} \alpha$ (b) $p^{2}=a^{2} \cos ^{2} \alpha+\sin ^{2} \alpha$ (c) $1 / p^{2}=\sin ^{2} \alpha+\alpha^{2} \cos ^{2} \alpha$ (d) $1 / p^{2}=\cos ^{2} \alpha+a^{2} \sin ^{2} \alpha$

If $x \cos \alpha+y \sin \alpha=p$ touches $x^{2}+a^{2} y^{2}=a^{2}$, then
(a) $p^{2}=a^{2} \sin ^{2} \alpha+\cos ^{2} \alpha$
(b) $p^{2}=a^{2} \cos ^{2} \alpha+\sin ^{2} \alpha$
(c) $1 / p^{2}=\sin ^{2} \alpha+\alpha^{2} \cos ^{2} \alpha$
(d) $1 / p^{2}=\cos ^{2} \alpha+a^{2} \sin ^{2} \alpha$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Tangency Condition for Conic Sections

A line touches a conic section if, when the line’s equation is substituted into the conic’s equation, the resulting quadratic in one variable has a single, repeated solution. This means that the discriminant of the quadratic must be zero. Establishing this condition allows one to derive relationships between the parameters of the line and the conic.

Ellipse and Its Standard Equation

An ellipse is a type of conic section that can be expressed in a standard form to explicitly show its axes and center. Understanding the standard form of an ellipse facilitates the analysis of its geometric properties, which is essential when determining the conditions under which a line is tangent to the ellipse.

Distance from a Point to a Line

The distance from a point to a line, calculated using the perpendicular distance formula, is a critical concept in geometric problems involving tangency. For a tangent line touching a conic, this distance from a reference point (often the center of the conic) typically aligns with a specific function of the conic’s parameters, thereby helping to establish the criterion for tangency.

Please give Ace some feedback