Shortest Distance on a Parabola: Point to Point Calculation

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Homework Help Overview

The problem involves finding the shortest distance from a point (1,4) to a point on the parabola defined by the equation y² = 2x. The discussion centers around methods for approaching this geometric problem within the context of calculus and optimization.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various steps to approach the problem, including implicit differentiation and finding tangents to the curve. There are questions about how to derive the tangent line and the application of the point-slope formula. Some participants suggest a minimization perspective, prompting a focus on distance formulas and variable relationships.

Discussion Status

The discussion is ongoing, with participants exploring different methods and expressing confusion about specific steps. Some guidance has been offered regarding the use of derivatives and distance formulas, but no consensus has been reached on a clear path forward.

Contextual Notes

Participants are navigating through the problem with varying levels of understanding, and there is mention of multiple potential approaches to the solution. The original poster's uncertainty about the next steps indicates a need for clarification on the mathematical concepts involved.

karisrou
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2. Find the shortest distance from the point (1,4) to a point on the parabola y^2 = 2x

Not really sure what to do here next. I'd imagine i might have to fuse implicit differentiation? But not really sure.
 
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1) solve for x
2) find tangent to curve at x
3) use point slope formula
4) solve for x

your ans should be sqrt(5)
 
I'm kind of confused on step 2. How do you find the tangent?
 
you know the x point so find its derivative and use the point slope formula and solve for x
 
I still don't really understand. Do you think you can show it?
 
There are multiple ways to solve this problem. One way is to think about this as an minimization problem. Hint: Let (x,y) be the point on the parabola closest to (1,4) and write out the formula for the distance between these two points. Next, use the relationship between x and y to write the formula in terms of one variable. Now what do you think you should do?
 

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