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Algebra Difficulty 3.0 AMC 10/12 Find the answer

In the rectangular coordinate system xOy, the parametric equation of line l is {y=tsinαx=tcosα\begin{cases} \overset{x=tcos\alpha }{y=tsin\alpha }\end{cases} (t is the parameter, α is the inclination angle), and the parametric equation of curve C is {y=2sinβx=4+2cosβ\begin{cases} \overset{x=4+2cos\beta }{y=2sin\beta }\end{cases} (β is the parameter, β∈0π0,π ). Establish a polar coordinate system with the coordinate origin O as the pole and the positive semi-axis of the x-axis as the polar axis.

(I) Write the ordinary equation of curve C and the polar coordinate equation of line l;

(II) If line l and curve C have exactly one common point P, find the polar coordinates of point P.

A number or a short expression. Spacing and $ signs are ignored.

Solution

(1) The parametric equation of curve C is {y=2sinβx=4+2cosβ\begin{cases} \overset{x=4+2cos\beta }{y=2sin\beta }\end{cases} (β is the parameter, β∈0π0,π ).

Converting to rectangular coordinate equation: (x-4)2+y2\=4 (y≥0).

The parametric equation of line l is {y=tsinαx=tcosα\begin{cases} \overset{x=tcos\alpha }{y=tsin\alpha }\end{cases} (t is the parameter, α is the inclination angle).

Converting to polar coordinate equation: θ=α.

(2) From (1), we know that curve C is a semicircle.

If line l and curve C have exactly one common point P, then line l is tangent to the semicircle.

Let P be (ρ, θ). From the problem, we know: sinθ=12sinθ= \frac {1}{2},

So: θ=π6θ= \frac {π}{6},

So: ρ2+22\=42,

Solving for ρ: ρ=23ρ=2 \sqrt {3}.

Therefore: Point P is (23,π6\boxed{2 \sqrt {3}, \frac {π}{6}}).

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.