It is given that the line x cosθ+y sinθ=p touches the curve ((x)/(6))^3 / 2+((y)/(4))^1 / 2=1 .

step 1 To solve the problem in which the equation of the curve is given as x over 6 risk to par 3 by 2

It is given that the line x cosθ+y sinθ=p touches the curve...

It is given that the line $x \cos \theta+y \sin \theta=p$ touches the curve $\left(\frac{x}{6}\right)^{3 / 2}+\left(\frac{y}{4}\right)^{1 / 2}=1 .$ Then (a) $(6 \cos \theta)^{3}+(4 \sin \theta)^{3}=\mathrm{p}^{3}$ (b) $(6 \cos \theta)^{3}-(4 \sin \theta)^{3}=\mathrm{p}^{3}$ (c) $(6 \cos \theta)^{3 / 2}+(4 \sin \theta)^{3 / 2}=\mathrm{p}^{3 / 2}$ (d) $36 \cos ^{2} \theta+16 \sin ^{2} \theta=\mathrm{p}^{2}$

It is given that the line $x \cos \theta+y \sin \theta=p$ touches the curve $\left(\frac{x}{6}\right)^{3 / 2}+\left(\frac{y}{4}\right)^{1 / 2}=1 .$ Then
(a) $(6 \cos \theta)^{3}+(4 \sin \theta)^{3}=\mathrm{p}^{3}$
(b) $(6 \cos \theta)^{3}-(4 \sin \theta)^{3}=\mathrm{p}^{3}$
(c) $(6 \cos \theta)^{3 / 2}+(4 \sin \theta)^{3 / 2}=\mathrm{p}^{3 / 2}$
(d) $36 \cos ^{2} \theta+16 \sin ^{2} \theta=\mathrm{p}^{2}$

Key Concepts

Normal form of a line

A line expressed in the normal form, x cos ? + y sin ? = p, represents the line in terms of its perpendicular distance from the origin and the angle that the perpendicular makes with the positive x?axis. This representation simplifies many geometric calculations, especially those involving distances and projections, and is widely used in problems relating to tangents, distances, and envelopes in analytic geometry.

Tangency condition for curves

The tangency condition is a criterion used to determine when a line touches a curve at exactly one point. It typically involves substituting the equation of the line into the equation of the curve to obtain an equation that must have a repeated (or double) root. This approach helps establish a relationship between the parameters defining the line and those of the curve, ensuring that the line is exactly tangent to the curve.

Elimination of parameters

Elimination of parameters involves substituting one relation into another to remove an independent variable or parameter, thereby deriving a condition or relation solely between the given constants. This technique is especially important in problems where a line is tangent to a curve because it allows the derivation of an explicit relationship between the parameters of the line (such as p and ?) and the properties of the curve.

Algebraic manipulation of fractional exponents

Many curves in analytic geometry are defined by equations involving fractional exponents, which require careful manipulation to simplify and derive useful relationships. Mastery of exponents’ rules and algebraic techniques is key to handling such expressions, making it possible to equate, combine, or transform terms when establishing conditions like tangency or when comparing different algebraic forms.

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