Find Min Distance b/w Parabolas: 7/(4*root2)

  • Thread starter Thread starter erisedk
  • Start date Start date
  • Tags Tags
    Minimum
Click For Summary
SUMMARY

The minimum distance between the curves defined by the equations y² - xy - 2x² = 0 and y² = x - 2 is calculated to be 7/(4√2). The first equation can be factored into two linear components, y = -x and y = 2x, simplifying the problem. By graphing these lines, it becomes evident that the line y = -x is closer to the curve y² = x - 2. The solution involves using the slope form of the tangent to a parabola to determine the distance between the two parallel lines.

PREREQUISITES
  • Understanding of parabolic equations and their properties
  • Knowledge of calculus, specifically the concept of tangents
  • Ability to graph linear equations and parabolas
  • Familiarity with distance formulas between lines
NEXT STEPS
  • Study the properties of parabolas and their equations
  • Learn about the slope form of the tangent to a parabola
  • Explore methods for finding distances between parallel lines
  • Practice factoring quadratic equations into linear components
USEFUL FOR

Students studying calculus, particularly those focusing on optimization problems involving parabolas and linear equations.

erisedk
Messages
372
Reaction score
7
No effort shown

Homework Statement


Find the minimum distance between the curves y^2 - xy - 2x^2 = 0 and y^2 = x - 2.
Ans: 7/(4*root2)

Homework Equations

The Attempt at a Solution


It's supposed to be done using concepts from parabolas. I tried using calc but it got convoluted.
 
Physics news on Phys.org
erisedk said:

Homework Statement


Find the minimum distance between the curves y^2 - xy - 2x^2 = 0 and y^2 = x - 2.
Ans: 7/(4*root2)

Homework Equations

The Attempt at a Solution


It's supposed to be done using concepts from parabolas. I tried using calc but it got convoluted.

y^2 - xy - 2x^2 splits into two linear factors. That makes it easier since the graph becomes two lines.
 
Ohhh ok. I didn't realize that was a pair of straight lines.
I did it now. Split it into y=-x and y=2x. Drew a rough graph. It's clear from the graph that y=-x is closer. So, using the slope form of tangent to a parabola, I wrote the equation of tangent with the slope -1, and figured out the distance between the two parallel lines, which comes out to be the answer. Thanks :D
 

Similar threads

Identifying the equation of a parabola
  • · Replies 2 ·
Replies
2
Views
2K
Find Minimum Value of T^2 in Normal to Parabola y^2 = 4ax at (at^2, 2at)
  • · Replies 3 ·
Replies
3
Views
3K
Finding the Parameters for the Curve y=asin(x-b)+c
  • · Replies 11 ·
Replies
11
Views
4K
Arriving at parabola formula via distance formula
  • · Replies 4 ·
Replies
4
Views
2K
Vector equation of a plane in normal form
  • · Replies 2 ·
Replies
2
Views
2K
What is the minimum distance between a tangent and normal of a parabola?
  • · Replies 15 ·
Replies
15
Views
5K
Finding the focus of a parabola given an equation
  • · Replies 4 ·
Replies
4
Views
2K
Finding the shortest distance from a point to the plane
  • · Replies 7 ·
Replies
7
Views
3K
Finding the Equation of a Parabola Tangent: Using Discriminant Method
  • · Replies 3 ·
Replies
3
Views
3K
Find the max and min value from equation?
  • · Replies 13 ·
Replies
13
Views
2K