The line x cosα+y sinα=P touches the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 if (a) P^2=a^2 cos^2 α+

step 1 In this question, if a variable line X -Pos alpha plus Y -sign alpha is equal to P, which is a c

The line x cosα+y sinα=P touches the hyperbola...

The line $x \cos \alpha+y \sin \alpha=P$ touches the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ if (a) $\mathrm{P}^{2}=\mathrm{a}^{2} \cos ^{2} \alpha+\mathrm{b}^{2} \sin ^{2} \alpha$ (b) $\mathrm{P}^{2}=\mathrm{b}^{2} \cos ^{2} \alpha+\mathrm{a}^{2} \sin ^{2} \alpha$ (c) $\mathrm{P}^{2}=\mathrm{a}^{2} \cos ^{2} \alpha-\mathrm{b}^{2} \sin ^{2} \alpha$ (d) $\mathrm{P}^{2}=\mathrm{b}^{2} \cos ^{2} \alpha-\mathrm{a}^{2} \operatorname{ain}^{2} \alpha$

The line $x \cos \alpha+y \sin \alpha=P$ touches the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ if
(a) $\mathrm{P}^{2}=\mathrm{a}^{2} \cos ^{2} \alpha+\mathrm{b}^{2} \sin ^{2} \alpha$
(b) $\mathrm{P}^{2}=\mathrm{b}^{2} \cos ^{2} \alpha+\mathrm{a}^{2} \sin ^{2} \alpha$
(c) $\mathrm{P}^{2}=\mathrm{a}^{2} \cos ^{2} \alpha-\mathrm{b}^{2} \sin ^{2} \alpha$
(d) $\mathrm{P}^{2}=\mathrm{b}^{2} \cos ^{2} \alpha-\mathrm{a}^{2} \operatorname{ain}^{2} \alpha$

Key Concepts

Optimization via Lagrange Multipliers in Geometric Contexts

The method of Lagrange multipliers is a technique used to find the extreme values (maximum or minimum) of a function subject to a constraint. In the context of conic sections, it can be applied to determine the extreme values of a linear function (such as a directional coordinate) on a conic. When the extreme value of the linear function coincides with a constant from the line's equation, the condition effectively defines a tangent line to the conic.

Line Representation in Normal Form

Expressing a line in the normal form, as x cos ? + y sin ? = P, means that the coefficients of x and y (namely cos ? and sin ?) form a unit normal vector. In this form, P represents the shortest distance from the line to the origin. This representation simplifies computations in analytic geometry, particularly when dealing with the distance from a point to a line or when applying conditions of tangency to conic sections.

Tangency Condition for Conic Sections

This concept refers to the requirement that a line touches a conic—such as a hyperbola, ellipse, or parabola—if and only if the system of equations representing the line and the conic yields a single intersection point. Algebraically, this is achieved by substituting the line’s equation into the conic’s equation, resulting in a quadratic equation whose discriminant must equal zero. This condition ensures that the line is tangent to the conic rather than secant (intersecting it at two points) or completely disjoint.

Standard Form and Properties of a Hyperbola

A hyperbola is a type of conic section defined by an equation of the form x²/a² - y²/b² = 1 (or a similar variant), which expresses the set of all points for which the difference of the distances to the two foci is constant. Understanding the standard form is crucial for analyzing geometric properties such as axes, asymptotes, and tangency conditions, all of which play an important role when determining whether a given line touches the hyperbola.

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