Parabola's tangent

[Loki123]

Write the equation of the tangent of the parabola y ^ 2 = 24x if it contains a point M (-1 / 6,0) that does not belong to the parabola.

The only difference between my answer and the book's answer is that I define k as a negative or a positive number. I am wrong?

 

[Dr.Peterson]

Write the equation of the tangent of the parabola y ^ 2 = 24x if it contains a point M (-1 / 6,0) that does not belong to the parabola.

The only difference between my answer and the book's answer is that I define k as a negative or a positive number. I am wrong? View attachment 31474

First, your answer is right, though it should be written as [imath]\pm(6x+1)[/imath] to make it clear that both signs must be the same. In the graph below, the solid red and green lines are your tangents, and the dotted lines are theirs:

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Clearly their lines don't both pass through the given point. So they've made a mistake, at least a typo.

As for your work, I don't know what you're doing. What does p mean, as you have been taught? Where does "p = 2kn" come from? (I would be using the derivative, but perhaps you haven't learned that, and this is a special formula you learned for the parabola.)

 

[Loki123]

First, your answer is right, though it should be written as [imath]\pm(6x+1)[/imath] to make it clear that both signs must be the same. In the graph below, the solid red and green lines are your tangents, and the dotted lines are theirs:

View attachment 31476​

Clearly their lines don't both pass through the given point. So they've made a mistake, at least a typo.

As for your work, I don't know what you're doing. What does p mean, as you have been taught? Where does "p = 2kn" come from? (I would be using the derivative, but perhaps you haven't learned that, and this is a special formula you learned for the parabola.)

P=2kn has to be true for y=kx+n to be a tangent to y^2=2px

 

[Dr.Peterson]

So you are just taught this as a formula with no further explanation, as I expected. I haven't see it in this form, but I find in on a couple sites that Google translates for me (from Serbian), that define p as the distance from the directrix to the focus, as I expected. (We usually use a parameter representing the distance from the vertex to the focus.) So everything makes sense.

Your work is correct.