Math Problem Statement
Solution
Let's analyze and work through this problem step-by-step.
Given:
- A line with the equation .
- A point on this line with abscissa .
We need to solve the following:
Part 1a: Show that
Solution Outline:
- Since is on the line , with abscissa , its coordinates are:
- The origin has coordinates .
- The square of the distance is given by:
- Substitute into this expression and expand it to obtain a form that matches the required result.
I'll work through these calculations now.
Part 1b: Find the value of that minimizes
Once we have the expression for in the required form, we will find the minimum by differentiating with respect to and setting the derivative to zero.
Part 2: Graph and Construct Point
For this part, we'll draw the line and locate the point at the minimum distance from .
Let me complete the calculations for Part 1a first.The expanded form of is:
To express this in the required form, , we can complete the square.
Let's proceed with completing the square for this expression.After completing the square, the expression becomes:
This matches the required form.
Part 1b: Find the value of that minimizes
To minimize , we set the derivative of with respect to to zero. Let's calculate this next.The value of that minimizes is:
Summary of Results
- We showed that .
- The value of that minimizes the distance is .
For part 2, you can now plot the line and locate the point with coordinates .
Would you like any further details or assistance with plotting, or have any additional questions?
Related Questions
- How do we verify the coordinates of point on the line ?
- What is the minimum distance when ?
- How can we interpret the completed square form of geometrically?
- Why does setting the derivative of to zero yield the minimum distance?
- How would the result change if the slope of the line were different?
Tip
Completing the square is a useful technique for converting quadratic expressions into a form that easily reveals minimum or maximum values.
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Distance Formula
Minimization
Algebraic Manipulation
Completing the Square
Formulas
Distance formula between two points: d^2 = (x2 - x1)^2 + (y2 - y1)^2
Completing the square to simplify quadratic expressions
Theorems
Concept of perpendicular distance from a point to a line
Suitable Grade Level
Grades 10-12