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Chapter 8: Problem 36

Find the angle between the curves \(2 y^{2}=x^{3}\) and \(y^{2}=32 x\).

Short Answer

Expert verified
The angles between the curves \(2y^2 = x^3\) and \(y^2 = 32x\) are given by \(胃 = atan\left(\frac{m2-m1}{1+m1*m2}\right)\), where \(m1\) and \(m2\) are the slopes of the tangent lines at \(y = -16, 0, 16\).

Step by step solution

01

Find the Points of Intersection

Set the two given equations equal to each other to find their intersection points: \(2y^2 = y^2 = 32x\). Then simplify: \(2y^2 = 32x\). Solving for \(x\), you get \(x = \frac{y^2}{16}\). Substituting this into the first equation gives \(\frac{2y^4}{256}=y^3\) which simplifies to \(y = -16, 0, 16\).
02

Find the Slopes of the Tangents

The slope of a curve at a given point is equal to the derivative of the curve鈥檚 function at that point. For \(2 y^2 = x^3\), the derivative is given by \( y'\ = \frac{3x^2}{4y}\). For \(y^2 = 32x\), the derivative is given by \(y' = \frac{1}{16}\frac{1}{2y}\), respectively. Evaluating these derivatives for \(y = -16\), \(y = 0\), and \(y = 16\) will give you the slopes of the tangent lines at the points of intersection.
03

Find the Angles

The arctangent of the difference between two slopes is equal to the angle between the two tangent lines and therefore between the curves. Calculate the angles using \(胃 = atan\left(\frac{m2-m1}{1+m1*m2}\right)\), where \(m1\) and \(m2\) are the slopes of the tangent lines. Remember, since each of these curves is symmetric, there will be two intersection points and thus two angles for \(y = -16\) and \(y = 16\) but not for \(y = 0\), as there is only one point of intersection there.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Lines
Understanding tangent lines is pivotal when studying the relationship between curves. A tangent line to a curve at a given point is a straight line that just touches the curve at that point. This line has the same direction as the curve at the point of tangency, which means it has the same slope as the curve at that point.

To find a tangent line's equation, you need one crucial piece of information: the slope of the curve at the point of tangency. This is found using derivatives, which leads us to the next important concept.
Derivatives
The derivative of a function at a point is the slope of the tangent line to the curve representing the function at that point. It indicates how the function is changing at that particular location. Calculating a derivative involves using calculus to find a formula for the slope of the tangent line at any point on the curve.

For instance, given the curve defined by the equation \(2y^2 = x^3\), the derivative, represented as \(y'\), helps us understand how quickly or slowly the curve is changing at any point \(x\). In the textbook example, the derivations lead to formulas \(y' = \frac{3x^2}{4y}\) for the first curve and \(y' = \frac{1}{16}\frac{1}{2y}\) for the second. These formulas can then be evaluated at specific points to find the angle between the curves.
Points of Intersection
Identifying the points of intersection between two curves is a crucial step in multiple mathematical and engineering applications. These points are where the two curves cross each other. Finding them typically involves setting the equations for the curves equal and solving for the variables.

In the exercise, setting the equations equal to each other and finding the common solutions unveils the x and y coordinates where the curves intersect. For our curves \(2y^2 = x^3\) and \(y^2 = 32x\), the intersection points play a fundamental role in determining where to evaluate the derivatives. Once these points are known, you can find the angle between the curves accurately by using the slopes of the tangent lines at these intersections.

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Most popular questions from this chapter

The point on the curve \(\sqrt{x}+\sqrt{y}=2 a^{2}\) at which the tangent is equally inclined to the axes is (a) \(\left(4 a^{4}, 0\right)\) (b) \(\left(0,4 a^{4}\right)\) (c) \(\left(a^{4}, a^{4}\right)\) (d) None

The equation to the normal to the curve \(y=\sin x\) at \((0,0)\) is (a) \(x=0\) (b) \(y=0\) (c) \(x+y=0\) (d) \(x-y=0\)

If the tangent at \(P\) to the curve \(y^{2}=x^{3}\) intersects the curve again at \(Q\) and the straight line \(O P, O Q\) makes angles \(\alpha, \beta\) with the \(x\)-axis, where \(O\) is origin, then find the value of \(\left(\frac{\tan \alpha}{\tan \beta}+2013\right)\).

Find the lengths of the tangent, sub-tangent, normal and sub-normal to the curve \(y^{2}=4 a x\) at the point \(P\left(\mathrm{at}^{2}, 2 \mathrm{at}\right)\).

The equation of the tangent at the point \(P(t)\), where \(t\) is any parameter, to the parabola \(y^{2}=5 a x\) is (a) \(y t=x+a t^{2}\) (b) \(y=x t+a t^{2}\) (c) \(y=t x\) (d) \(y=x+\frac{a}{t}\)
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