Proving Closest Point on Circle to (3,4): Calculus Solution

[science.girl]

Homework Statement

Show that the point (3/5, 4/5) is the closest point on the circle x^{2} + y^{2} = 1 to the point (3, 4).

Homework Equations

N/A

The Attempt at a Solution

The only equation that comes to mind is the distance formula... but I don't know what I would show with that. I know the solution deals with calculus, I just don't know what concepts to apply.

 

[JG89]

Why don't you write down the distance formula for any point on the circle to the point (3,4), and show that the minimum occurs at the point (3/5, 4/5)?

 

[science.girl]

Why don't you write down the distance formula for any point on the circle to the point (3,4), and show that the minimum occurs at the point (3/5, 4/5)?

Ah, that makes sense. One question, though... do I have to solve the original equation for x₁ and y₁ before substituting into the distance equation? (I just don't see how you set up the circle equation and various points via the distance equation.) From there, though, I know how to find the minimum.

 

[HallsofIvy]

You don't have to solve for y- you can use "inplicit differentiation". But you don't have to differentiate at all:

One crucial point about "min distance" is that the line from a point to a curve that is of minimum distance is perpendicular to the (tangent to the) curve. And, since this is a circle, a tangent is always perpendicular to the radius. The minimum distance from a point to a circle is always measured along the extended radius of the circle.

Since a radius goes through the center, what is the equation of the line through the center of this circle and (3,4)? where does that line intersect the circle? (There will be two points where that line intersects the circle, of course, but one is obviously "nearest", the other "farthest" from the point.)