AskSia

Plus

22. 在平面直角坐标系

x O y

中, 以坐标原点

O

为极点,

x

轴的正半轴为䢁轴建立极坐标系, 曲线

C

的极坐标方程为 $\rho...

Aug 23, 2024

Solution by Steps

step 1

To convert the polar equation

\rho = \rho \cos \theta + 1

to rectangular coordinates, we start with the relationship between polar and Cartesian coordinates:

\rho = \sqrt{x^2 + y^2}

and

\cos \theta = \frac{x}{\rho}

. Substituting these into the equation gives us:

\sqrt{x^2 + y^2} = \sqrt{x^2 + y^2} \cdot \frac{x}{\sqrt{x^2 + y^2}} + 1

step 2

Simplifying the equation from step 1, we have:

\sqrt{x^2 + y^2} - x = 1

. Squaring both sides results in:

x^2 + y^2 - 2x + x^2 = 1

, which simplifies to

2x^2 + y^2 - 2x - 1 = 0

step 3

Rearranging the equation gives us:

y^2 = 2x - 2x^2 + 1

. This can be expressed as

x = \frac{y^2}{2} - \frac{1}{2}

step 4

For the line

l: \begin{cases} x = t \\ y = t + a \end{cases}

, we need to find the points of intersection with the curve

C

. The distance

|AB| = 2

implies that the difference in

y

-coordinates of points

A

and

B

must equal 2

step 5

Setting up the equation for the distance between points

A

and

B

gives us

|(t + a) - (t + a + 2)| = 2

. Solving this leads to

|2| = 2

, confirming the distance condition

step 6

Since the variable

a

does not appear in the distance equation, we conclude that

a

can take any value that satisfies the intersection condition. However, we need to find a specific value of

a

that maintains the distance

|AB| = 2

Answer

The value of

a

can be determined based on the specific intersection points, but it is not uniquely defined from the given conditions.

Key Concept

The conversion from polar to rectangular coordinates and the analysis of intersection points.

Explanation

The process involves transforming the polar equation into Cartesian form and analyzing the conditions for intersection with a line, leading to a general conclusion about the variable

a