# Find the shortest distance between the line and the curve?

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#y-x=1# and #x=y^2#

##### 1 Answer

#### Explanation:

Here's a method that does not use differentiation.

Given:

#{ (y - x = 1), (x = y^2) :}#

The graphs of these equations look something like this:

graph{(y-x-1)(x-y^2) = 0 [-5, 5, -2.5, 2.5]}

Let's find a line parallel to

Given a system of equations:

#{ (y - x = k), (x = y^2) :}#

we want to find the value of

Substituting

#y - k = y^2#

That is:

#y^2 - y + k = 0#

This is a quadratic in standard form:

#ay^2+by+c = 0#

with

This has discriminant

#Delta = b^2-4ac = (-1)^2-4(1)(k) = 1-4k#

So this quadratic has exactly one root when

So the parabola

graph{(y-x-1)(y-x-1/4)(x-0.02-y^2) = 0 [-2.5, 2.5, -1.25, 1.25]}

Since the lines are diagonal, the distance between them is:

#(1-1/4)sqrt(2)/2 = (3sqrt(2))/8#