https://r2.erweima.ai/mathbot/image/1725763909214hhvq3kwx.jpg

The problem shown in the image involves quadratic functions and geometric interpretations. The task seems to include several parts:

### Problem Breakdown:
1. **Intersection of Curves**: The first part asks for the condition under which the quadratic function \( y = \frac{1}{2} x^2 + mx + 1 \) intersects with the line \( y = \frac{1}{x} \) at point \( B(3,1) \).
2. **Determining Parameters**: The second part requires finding the value of \( m \) for the intersection.
3. **Angle of Intersection**: The third part seems to involve the calculation of the angle between the tangent at the point \( B \) and the x-axis.
4. **Geometric Properties**: The fourth part involves a geometric interpretation possibly related to distances and certain properties of triangles or angles.

### Step-by-Step Analysis:

#### 1. **Finding the Value of \( m \)**:
   To find the intersection at \( B(3, 1) \), you substitute \( x = 3 \) and \( y = 1 \) into the equation of the quadratic function:
   \[
   1 = \frac{1}{2}(3)^2 + 3m + 1
   \]
   Simplifying:
   \[
   1 = \frac{9}{2} + 3m + 1 \quad \Rightarrow \quad 3m = 1 - \frac{9}{2} - 1 = -\frac{11}{2} \quad \Rightarrow \quad m = -\frac{11}{6}
   \]

#### 2. **Finding the Condition**:
   This involves ensuring the quadratic and the line intersect at point \( B(3, 1) \) by solving the equations simultaneously.

#### 3. **Angle of Intersection**:
   The angle between the tangent to the curve and the x-axis can be found using the derivative of the quadratic function at \( x = 3 \):
   \[
   y' = x + m
   \]
   At \( x = 3 \), the slope of the tangent \( y' = 3 + m = 3 - \frac{11}{6} = \frac{7}{6} \).

#### 4. **Further Geometric Interpretation**:
   The problem might involve specific geometric properties like distances between points or specific angles in triangles formed by the intersection points.

If you need further details or clarifications on any part of the solution, feel free to ask! 

### Related Questions:
1. What is the significance of the point \( B(3,1) \) in the context of the intersection?
2. How can you generalize the method to find the intersection of any quadratic and linear function?
3. What does the slope of the tangent line tell us about the curvature of the parabola?
4. How would the solution change if the quadratic coefficient were different?
5. What geometric properties are key when analyzing intersections of curves?

**Tip:** Understanding the relationship between algebraic conditions (like finding the intersection) and geometric interpretations (like angles between curves) is crucial in tackling problems involving curves and lines.

Determine the conditions for the quadratic curve y = (1/2)x^2 + mx + 1 to intersect with the curve y = 1/x at point B(3,1).

Analyze the intersection of the quadratic function y = (1/2)x^2 + mx + 1 with the curve y = 1/x at point B(3,1). This problem covers key concepts like quadratic equations, derivatives, and geometric interpretation of curve intersections.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation