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