SOLUTION: What is the equation formed from the families of straight lines tangent to the parabola {{{y^2=2x}}}

Question 1151870: What is the equation formed from the families of straight lines tangent to the parabola
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!

the equation of tangent line that we seek is
if , then
Since both the abscissa () and ordinate () are positive, we can try to work with the positive half of the function when solving for :

To find the line tangent to it, we find the derivative

Plug in to get the slope

To get the y-intercept, plug in the numbers in the slope-intercept form linear equation

when =>=>, and tangent will touch parabola at point (,)
we have
=>
=>
=>
So the equation is

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The graph of is symmetrical with respect to the x-axis. So we can find the equations of the tangent lines for the portion of the graph with y positive; then any tangent line with equation will have a corresponding tangent line with equation , or .

The equation for the portion of the graph with y positive is

Take the derivative, using the power and chain rules, to find the slope:

For each value of x, the tangent line will have slope ; and the y value will be

For a given value of x, find the expression for the y-intercept b for the tangent line with and slope .

So for a given value of x, the tangent line in the first quadrant has slope and y-intercept .

The equation is then

Choosing some "nice" values of x that make and rational....

x = 1/2:
x = 2:
x = 9/2:
x = 8:

And then there are corresponding tangent lines on the other branch of the parabola:

x = 1/2:
x = 2:
x = 9/2:
x = 8:

A graph of the parabola and the tangent lines for x=1/2 -- x+1/2 and -x-1/2:

A graph of the parabola and the tangent lines for x=2 -- (1/2)x+1 and (-1/2)x-1:

A graph of the parabola and the tangent lines for x=8 -- (1/4)x+2 and (-1/4)x-2:

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