What is the condition that the line lx+my+n=0 is tangent to parabola y^2=4a(x-a)?

Answer 1

lx + my + n is tangent to y^2 = 4a(x-a)

Careful reading -- letter l, not number 1.

This parabola is "sideways" compared to our more common y = x^2 parabola.For any given value of x, there are two y values (this is not a function). If a is positive it opens to the right, and if a is negative it opens to the left. Vertex (see worked out below) is (a,0)

We need to calculate the slope of the tangent to that parabola. If you have not done calculus yet, you can do that by taking the limit of the secant. Using calculus with *implicit differentiation* (derivative of both sides and *chain rule* on any function of y rather than x)

y^2 = 4a(x-a)

y^2 = 4ax - 4a^2

2y * dy/dx = 4a [ d/dx both sides; chain rule on y^2; derivative of constant 4a^2 is zero]

I use the dy/dx form for many reasons including here the fact that it makes it clear who the independent variable is. dy/dx = y' IF we are sure x is the independent variable.

If y is not zero,

dy/dx = 4a/2y

dy/dx = 2a/y

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If y = 0, then 0 = 4a(x-a)

Given a non-zero, y = 0 at x = a and this is the vertex of the sideways parabola. If you sketch it you will see at this point the tangent is vertical, slope undefined (infinite)

At the vertex (a,0) the tangent is x = a

Compare to l x + my + n = 0,

we get m = 0, la + n = 0, n = -la

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Now we look at the line

lx + my + n = 0

my = -lx - n

**IF** m is not zero

y = (-l/m) x - n/m

The slope is -l/m and the y-intercept is -n/m

So n/m = (-l/m)x - y

So the line is tangent to the parabola if 2a/y = -l/m

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We are free to choose any real y value for the tangent. at the point ( (y^2/)4a + a, y)

OR equivalently free to choose any x such that x >= a [given a positive; reverse if a is negative] for the tangent at the point (x, plus or minus 2 sqrt(x(x-a))

We are also free to choose any non-zero m. since the line is the same when multiplied by any constant

Then l/m = -2a/y and n/m = (-l/m)x - y

So choose either y or x as above, choose an m

[We could logically choose any one of l, m, or n and then find the restrictions on the others; this seemed easiest]

--> If y = 0, x = a, m = 0, and n = - la [vertical line]

--> if y not equal zero and x not equal a, l = (-2a/y)*m and n = -lx - my