Parabola tangent

[Loki123]

Write the equation of the tangent of the parabola y ^ 2 = 16x if it contains a point M (0,-1/2) that does not belong to the parabola.
My answer is way off.

 

[The Highlander]

Write the equation of the tangent of the parabola y ^ 2 = 16x if it contains a point M (0,-1/2) that does not belong to the parabola.
My answer is way off. View attachment 31475

The tangent is a straight line (y=mx+c) which shares a single point with the parabola (y²=16x) and you are given only the y-intercept (c=-½).

Your approach appears to assume that the tangent will pass through the point (16,-64), ie: where there is an ordinate at 2p (p=8) which seems to be based on the various analyses of your first question?

However, that would give an equation for the tangent of y=-3.96875x-½ (m=63.5/16) but that line crosses the parabola at two points (see attached) and is, therefore not a tangent!

Perhaps you should be considering the distance between the point (0,-½) and some point (x, y) that satisfies the parabolic equation (y²=16x) and trying to find a minimum value for that distance?

 

[The Highlander]

By the way, wtf is this "BOOK"?
Please tell us more about it (and what language your copy is printed in).

 

[The Highlander]

Oops!
There was a slight boob in my arithmetic above, it should have said:-
"the tangent will pass through the point (16,-16)"
and
"an equation for the tangent of y=-0.96875x-½ (m=15.5/16)"
However, I trust that would not affect any of the rest of what I said?

 

[Dr.Peterson]

Write the equation of the tangent of the parabola y ^ 2 = 16x if it contains a point M (0,-1/2) that does not belong to the parabola.
My answer is way off. View attachment 31475

Since the book's

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isn't even the equation of a line, you must be looking at the wrong answer. Don't assume your answer is wrong!

Your equation is one of the two tangent lines through the given point, the other being the vertical line x = 0! (The latter can't be found by your method, because the slope is undefined.)

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By the way, wtf is this "BOOK"?
Please tell us more about it (and what language your copy is printed in).

Please be polite. I have discovered that the formulas used here are found in Serbian sources, so I presume that is where this is from.

Other answers from the book have been correct, so I assume this is an anomaly. I don't understand your analysis at all.

 

[Dr.Peterson]

I just tried graphing [imath]4x\pm\sqrt{2}y+2=0[/imath], pulling the y outside of the radical where you had written it, and saw that these are tangents through [imath](-\frac{1}{2},0)[/imath], rather than [imath](0,-\frac{1}{2})[/imath]:

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So either you or they misread the problem! And you solved either the right problem or the wrong problem, in either case correctly.