Math Problem Statement

Determine all points on line L that are twice as far from (1,3) as they are from (7,5). The problem involves finding points p and q such that the distance from p to (1,3) is twice the distance from p to (7,5), and similarly for q.

Solution

The exercise involves determining all points on the line LL that are twice as far from (1,3)(1,3) as from (7,5)(7,5). The problem is asking for points pp and qq that satisfy this condition, and the goal is to calculate vectors VV and WW, as well as to determine the scaling factors involved.

Let's break down the key steps:

  1. Sketch the vectors VV and WW:

    • Vector V=(7,5)pV = (7,5) - p, which represents the vector from pp to (7,5)(7,5).
    • Vector W=q(7,5)W = q - (7,5), which represents the vector from (7,5)(7,5) to qq.
  2. Determine scaling factors:

    • The problem states that the distance between pp and (1,3)(1,3) is twice that between pp and (7,5)(7,5).
    • Likewise, you need to find a similar relationship for the vector involving qq.

For each part of the problem:

  • Part (a): You should graphically represent the vectors.
  • Part (b): The vectors VV and WW are described as multiples of (7,5)(1,3)(7,5) - (1,3), so you need to determine their relationship.
  • Part (c): Compute the coordinates of points pp and qq by solving the distance ratio equations for the given conditions.

Would you like a detailed breakdown of each part with corresponding calculations?


Here are 5 questions to deepen understanding:

  1. How do we compute the distance between two points in a plane?
  2. What is the relationship between vector scaling and distances?
  3. How can you express the condition of "twice as far" using vector norms?
  4. What kind of equation would you set up to find points equidistant from two points?
  5. What geometric shape describes the locus of points equidistant from two given points?

Tip: When working with vectors, always remember that the magnitude of a vector represents the distance between the two points involved.

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Math Problem Analysis

Mathematical Concepts

Vector Analysis
Distance Formula
Ratio and Proportion
Linear Equations

Formulas

Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Vector representation: V = (7,5) - p and W = q - (7,5)

Theorems

Distance-Point Ratio Theorem
Properties of Vectors and Magnitudes

Suitable Grade Level

Grades 10-12