Understanding the Acute Angle of Intersection Between Two Curves
In this article, we will explore the acute angle of intersection between the curves given by the equations (y sqrt{4x - x^2}) and (y sqrt{8 - x^2}). The term "acute angle" refers to the smallest angle formed by the intersection of the two curves. This angle can be found by analyzing the tangent lines at the points of intersection.
Identifying the Intersection Points
To find the points of intersection, we need to set the two equations equal to each other:
[sqrt{4x - x^2} sqrt{8 - x^2}]Squaring both sides of the equation to eliminate the square roots gives:
[4x - x^2 8 - x^2]After simplifying, the equation reduces to:
[4x 8]Solving for (x) yields:
[x 2]Substituting (x 2) back into either of the original equations, we find:
[y sqrt{4(2) - 2^2} sqrt{8 - 4} sqrt{4} 2]Thus, the point of intersection is ((2, 2)).
Determining the Slopes of the Tangent Lines
To find the angle of intersection, we need to determine the slopes of the tangent lines to the curves at the point of intersection. The slope of the tangent line to a curve is given by the derivative of the curve at that point.
Slopes of the Tangent Lines
Curve (y sqrt{4x - x^2})
The derivative of (y sqrt{4x - x^2}) is:
[y' frac{4 - 2x}{2sqrt{4x - x^2}} frac{2 - x}{sqrt{4x - x^2}}]At ((2, 2)), the slope is:
[m_1 frac{2 - 2}{sqrt{4(2) - 2^2}} 0]The tangent line at this point is vertical, and its equation is:
[y 2]Curve (y sqrt{8 - x^2})
The derivative of (y sqrt{8 - x^2}) is:
[y' frac{-x}{sqrt{8 - x^2}}]At ((2, 2)), the slope is:
[m_2 frac{-2}{sqrt{8 - 2^2}} frac{-2}{sqrt{4}} -1]The tangent line at this point has a slope of -1, and its equation is:
[y - 2 -1(x - 2)] [y 4 - x]Calculating the Angles of Intersection
The angle (theta_1) that the vertical tangent line makes with the positive x-axis is:
[theta_1 arctan(0) 0]The angle (theta_2) that the tangent line (y 4 - x) makes with the positive x-axis is:
[theta_2 arctan(-1) frac{3pi}{4}]The difference in angles is:
[theta_2 - theta_1 frac{3pi}{4} - 0 frac{3pi}{4}]The problem asks for the acute angle, which is the smaller of the two angles formed by the intersection. Therefore, the acute angle is:
[pi - frac{3pi}{4} frac{pi}{4}]Thus, the acute angle of intersection is (45) degrees.
Conclusion
In conclusion, the acute angle of intersection between the curves (y sqrt{4x - x^2}) and (y sqrt{8 - x^2}) is (45) degrees. This angle is determined by finding the slopes of the tangent lines at the point of intersection and then calculating the difference in the angles these slopes make with the x-axis.