If θ denotes the acute angle between the curves, y = 10 - x² and y = 2 + x² at a point of their inte

Q: Why do we use the absolute value in the formula?

The angle between two curves is defined as the acute angle. Since $\tan \theta$ can be negative for obtuse angles, the absolute value ensures we get the positive value corresponding to the acute angle.

Q: Does it matter which intersection point I choose?

No, due to the symmetry of the parabolas, the angle at $x=2$ is the same as the angle at $x=-2$. You can pick either one.

Q: What if the denominator $1 + m_1 m_2$ is zero?

If the denominator is zero, the tangents are perpendicular, and the angle is $90^\circ$. In this case, $\tan \theta$ would be undefined, which is a standard result for perpendicular lines.

Question

If $	heta$ denotes the acute angle between the curves, $y = 10 - x^2$ and $y = 2 + x^2$ at a point of their intersection, then $|	an 	heta|$ is equal to :

Options:

(A) 4/9

(B) 7/17

(C) 8/17

(D) 8/15

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine standing on a coordinate plane, watching two paths cross. One path is a downward-opening parabola, $y = 10 - x^2$, and the other is an upward-opening parabola, $y = 2 + x^2$.

To find the angle between them, we must first identify their points of intersection. By setting the two equations equal:

$$10 - x^2 = 2 + x^2$$

This simplifies to $2x^2 = 8$, which further reduces to $x^2 = 4$. This yields two points of intersection at $x = 2$ and $x = -2$.

Focusing our attention on the point where $x = 2$, we substitute this back into our equations to find $y = 6$. Thus, our intersection point is $P(2, 6)$.

### The Calculus of Tangents

The angle between two curves is defined as the angle between their tangent lines at the point of intersection. To find these tangents, we utilize the power of calculus.

For the first curve, $C_1: y = 10 - x^2$, the derivative is:

$$\frac{dy}{dx} = -2x$$

Evaluating this at $x = 2$, we obtain the slope $m_1 = -2(2) = -4$.

For the second curve, $C_2: y = 2 + x^2$, the derivative is:

$$\frac{dy}{dx} = 2x$$

Evaluating this at $x = 2$, we obtain the slope $m_2 = 2(2) = 4$. We now have two lines with slopes $m_1 = -4$ and $m_2 = 4$ passing through the same point.

### The Final Calculation

The angle $	heta$ between two lines with slopes $m_1$ and $m_2$ is given by the formula:

$$	an 	heta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} ight|$$

Substituting our calculated values into the formula:

$$	an 	heta = \left| \frac{-4 - 4}{1 + (-4)(4)} ight|$$

This simplifies to:

$$	an 	heta = \left| \frac{-8}{1 - 16} ight| = \left| \frac{-8}{-15} ight|$$

The negative signs cancel out, leaving us with the final result:

$$	an 	heta = \frac{8}{15}$$

This result is a testament to how calculus allows us to quantify the geometry of curves. You have successfully navigated the intersection, calculated the slopes, and applied the angle formula to find the **exact value of $	an 	heta = \frac{8}{15}$**.

Animated Solution for Mathematics - Differentiation: If $θ$ denotes the acute angle between the curves, $y = 10 - x^{2}$ and $y = 2 + x^{2}$ at a point of their intersection, then $∣ tan θ ∣$ is equal to :

Select Answer:

Visualized Solution

Visualize the Curves

Finding the Intersection Point

Solving for $x$

Finding the $y$ -coordinate

Concept: Angle Between Curves

Slope of Tangent to $C_{1}$ ( $m_{1}$ )

Slope of Tangent to $C_{2}$ ( $m_{2}$ )

The Angle Formula

Substituting the Slopes

Calculating the Value

Final Answer

The Sigma Insight: Tangents, Normals and Rate Measure

Analyzing the Setup

The Calculus of Tangents

The Final Calculation

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