This kind of question long time no do, very rusty liao.
After some thought, my working goes like this:
Note y^2 = 4x
x = (1/4)y^2
And y = (x/k) + k
x = ky - k^2 (multiply by k throughout)
Now treat x as your typical y-coordinate, y as your typical x-coordinate. That is, switch the axis.
Rewrite the equations, we have:
y = (1/4)x^2 and y = kx - k^2
By showing y = kx - k^2 is a tangent to the curve y = (1/4)x^2 for all values of k, we would have solved the question itself. Understand?
From y = (1/4)x^2, differentiate w.r.t x
dy/dx = (1/2)x
Recall your tangent to the curve is always in the standard form y = mx + c, where m is actually your dy/dx.
To prevent confusion, we will use Y = mX + c, so now we have
Y = [(1/2)x]X + c (Note, x and X are different variables) ----- (1)
Now we need to find what is c. By an "educated guess", find the value of Y when X = x
From Y = (1/4)X^2 (Equation of curve), we have Y = (1/4)x^2
Sub into (1): (1/4)x^2 = [(1/2)x]x + c
(1/4)x^2 = (1/2)x^2 + c
c = - (1/4)x^2
So, you will have actually find out the general form of the tangent to the curve
Y = (1/4)X^2, which is Y = [(1/2)x]X - (1/4)x^2. Observe this works for all values of x.
Hence, by letting k = (1/2)x, you will already have your y = kx - k^2, which is what you want to show. This thus works for all values of k.
As a side note, you shall put that k = 0 is not included since anything over 0 is an undefined value. But since the question says for all values of k, I think it's alright not to include it.
