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已知曲线

C_{1} 、 C_{2}

分别是函数

y=-\frac{4}{x}(x<0), y=\frac{k}{x}(k>0, x>0)

的图像, 边...

Mar 9, 2024

已知曲线

C_{1} 、 C_{2}

分别是函数 y=-\frac{4}{x}(x<0), y=\frac{k}{x}(k>0, x>0) 的图像, 边长为 6 的正

\triangle A B C

的顶点

A

在

y

轴正半轴上, 顶点

B 、 C

在

x

轴上 (

B

在

C

的左侧) , 现将

\triangle A B C

绕原点

O

函时针筂转, 当点

B

在曲线

C_{1}

上时, 点

A

恰好在曲线

C_{2}

上, 则k的值为

Solution by Steps

step 1

Identify the coordinates of vertices A, B, and C

step 2

Since vertex A is on the positive y-axis and the triangle is equilateral with side length 6, the coordinates of A are (0, 6)

step 3

Vertices B and C are on the x-axis, so their y-coordinates are 0. The x-coordinate of B is negative, and C is positive

step 4

The distance between B and C is the side length of the triangle, which is 6. So, if B is at (-3, 0), then C is at (3, 0)

step 5

When the triangle is rotated, vertex B will lie on curve

C_1

and vertex A will lie on curve

C_2

step 6

The equation of curve

C_1

y = -\frac{4}{x}

for x < 0

step 7

The equation of curve

C_2

y = \frac{k}{x}

for k > 0 and x > 0

step 8

Since B lies on

C_1

, we substitute the coordinates of B into the equation of

C_1

to find the value of y when x is -3

step 9

Substituting B into

C_1

gives

y = -\frac{4}{-3} = \frac{4}{3}

step 10

Since A lies on

C_2

, we substitute the coordinates of A into the equation of

C_2

to find the value of k when x is 3

step 11

Substituting A into

C_2

gives

6 = \frac{k}{3}

, which implies

k = 18

Answer

k = 18

Key Concept

Finding the value of k for a rotated equilateral triangle with vertices on given curves

Explanation

By placing the vertices of the equilateral triangle on the given curves and using the side length, we can solve for the unknown value k in the equation of curve

C_2