Here is a detailed and educational solution to the problem, designed for a JEE aspirant.

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Understanding the Angle Between Curves: The Core Concept

In differential calculus, the angle between two curves at a point of intersection is defined as the angle between their respective tangent lines at that specific point. This is a fundamental concept frequently tested in JEE.

If two lines have slopes $m_1$ and $m_2$, the angle $\theta$ between them is given by the formula: $\tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2}$ For the acute angle between the curves, which is typically what problems ask for unless specified otherwise, we take the absolute value of this expression: $|\tan \theta| = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$ The absolute value ensures that $\tan \theta$ is positive, corresponding to an acute angle ($0 < \theta \le \pi/2$).

Special Case: If the denominator $1 + m_1 m_2 = 0$, it implies that $m_1 m_2 = -1$. This condition means the tangent lines are perpendicular, and thus the angle between the curves is $90^\circ$ (or $\pi/2$ radians). In this scenario, $\tan \theta$ would be undefined.

Now, let's systematically apply this concept to the given problem.

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Step 1: Finding the Points of Intersection of the Curves

The very first step in determining the angle between two curves is to find where they actually meet. The angle between them is defined only at these common points.

The given curves are:

1. $y = 10 - x^2$ (a downward-opening parabola)
2. $y = 2 + x^2$ (an upward-opening parabola)

Why this step? We need the specific coordinates $(x, y)$ of the intersection points because the slopes of the tangent lines (which we'll find using derivatives) depend on the $x$-coordinate. The angle between the curves is specific to these points, not a general property.

To find the intersection points, we set the $y$-values equal: $10 - x^2 = 2 + x^2$ Now, we solve for $x$: $10 - 2 = x^2 + x^2$ $8 = 2x^2$ $x^2 = 4$ Taking the square root of both sides gives: $x = \pm 2$ These are the $x$-coordinates where the curves intersect.

Next, we find the corresponding $y$-coordinates by substituting these $x$ values into either of the original equations. Let's use the second equation, $y = 2 + x^2$, as it's slightly simpler:

• For $x = 2$: $y = 2 + (2)^2 = 2 + 4 = 6$
• For $x = -2$: $y = 2 + (-2)^2 = 2 + 4 = 6$

So, the two points of intersection are $(2, 6)$ and $(-2, 6)$. Due to the symmetry of both parabolas about the y-axis, the acute angle between the curves will be the same at both intersection points. We can choose either point for our calculations. Let's proceed with the point $(2, 6)$.

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Step 2: Determining the Slopes of the Tangent Lines (Using Differentiation)

To find the slopes ($m_1$ and $m_2$) of the tangent lines to each curve at an intersection point, we need to use differential calculus. The derivative $\frac{dy}{dx}$ of a curve's equation gives the general formula for the slope of its tangent at any point $(x, y)$ on the curve.

Why this step? The formula for $\tan \theta$ directly requires the slopes of the tangent lines. Differentiation is the mathematical tool to obtain these general slope expressions from the curve equations.

Let's find the derivatives for both curves:

For Curve 1: $y = 10 - x^2$ Differentiating with respect to $x$: $\frac{dy}{dx} \Big|_1 = \frac{d}{dx}(10 - x^2) = 0 - 2x = -2x$

For Curve 2: $y = 2 + x^2$ Differentiating with respect to $x$: $\frac{dy}{dx} \Big|_2 = \frac{d}{dx}(2 + x^2) = 0 + 2x = 2x$

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Step 3: Evaluating Slopes at the Specific Intersection Point

The derivatives we just found give us the general slope of the tangent at any $x$ on each curve. Now, we need to find the specific numerical slopes of the tangent lines at our chosen point of intersection, $(2, 6)$.

Why this step? The angle between the curves is measured at a particular point. Therefore, we must evaluate the general slope formulas (the derivatives) at the coordinates of that specific point to get the actual numerical slopes $m_1$ and $m_2$ that will be used in the angle formula.

Let $m_1$ be the slope of the tangent to Curve 1 at $(2, 6)$: $m_1 = \left( \frac{dy}{dx} \Big|_1 \right)_{x=2} = -2(2) = -4$

Let $m_2$ be the slope of the tangent to Curve 2 at $(2, 6)$: $m_2 = \left( \frac{dy}{dx} \Big|_2 \right)_{x=2} = 2(2) = 4$

So, we have the two slopes at the point $(2, 6)$: $m_1 = -4$ and $m_2 = 4$.

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Step 4: Calculating the Tangent of the Acute Angle

Now that we have the slopes $m_1$ and $m_2$ at the intersection point, we can use the formula for the tangent of the acute angle between the tangent lines (and thus between the curves).

Why this step? This is the final calculation to directly obtain the required value, $|\tan \theta|$, using the numerical slopes we just found.

Using the formula for the acute angle: $|\tan \theta| = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$ Substitute the values $m_1 = -4$ and $m_2 = 4$: $|\tan \theta| = \left| \frac{(-4) - (4)}{1 + (-4)(4)} \right|$ $|\tan \theta| = \left| \frac{-8}{1 - 16} \right|$ $|\tan \theta| = \left| \frac{-8}{-15} \right|$ $|\tan \theta| = \left| \frac{8}{15} \right|$ Since $\frac{8}{15}$ is already a positive value, the absolute value does not change it: $|\tan \theta| = \frac{8}{15}$

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Important Tips for Success and Common Pitfalls

• Order of Operations is Crucial: Always follow a systematic approach:
  1. Find the points of intersection first.
  2. Then, differentiate each curve equation to find the general slope formula.
  3. Finally, evaluate the derivatives at those specific intersection points to get the numerical slopes. Don't mix up this order!
• Evaluate at the Correct Point: Remember that $\frac{dy}{dx}$ gives a general slope formula. You must substitute the $x$-coordinate of the intersection point into the derivative to get the specific numerical slope for that point. A common mistake is to use the general derivative directly without evaluating.
• Acute Angle: The question explicitly asks for the acute angle, indicated by $|\tan \theta|$. Always use the absolute value in the formula unless specifically asked for the non-acute angle.
• Perpendicular Curves: Be mindful of the denominator $1 + m_1 m_2$. If it becomes zero, the curves intersect orthogonally (at $90^\circ$). In such a case, $\tan \theta$ would be undefined, and the answer would be $\theta = \pi/2$.
• Calculus Fundamentals: Ensure you are comfortable with basic differentiation rules (power rule, constant rule) as they are frequently used in such problems.
• Algebraic Accuracy: A small error in solving for $x$, in differentiation, or in substituting values can lead to an incorrect final answer. Double-check your arithmetic carefully.

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Summary and Key Takeaway

To find the angle between two curves, the systematic and reliable approach is:

1. Find Intersection Points: Equate the curve equations and solve for the coordinates $(x, y)$ where they meet.
2. Find General Slopes: Differentiate each curve equation with respect to $x$ to get the general expressions for $\frac{dy}{dx}$.
3. Find Specific Slopes: Evaluate the general slope expressions for each curve at the chosen intersection point to obtain the numerical slopes $m_1$ and $m_2$.
4. Apply Angle Formula: Use the formula $|\tan \theta| = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$ to calculate the tangent of the acute angle.

Following these steps meticulously ensures you arrive at the correct solution.

The final answer is $\boxed{\frac{8}{15}}$.