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How to show a line is the tangent in a curve?

Hey so i have just done a solomon paper where the question was

show that the line y=8-2x is a tangent to the curve y=x²-6x+12?

I basically equated the two gradients together e.g. -2=2x-6 to say that the line must be a tangent when x =2 but am i allowed to do this?

In the mark scheme it equated the to equations together to say that as (x-2)^2 is a repeated root, it must be a tangent? Could someone also explain this to me?

Thanks x

Reply 1

the bear

you do what you said to make a new quadratic equation. if this has a repeated root then the line must be a tangent to the curve.

Reply 2

Tubbz

Yeah you basically prove a point is on both the curve and the line which means they touch at that point, then prove the line and curve have the same gradients at that point.

Reply 3

Appazap

Original post by the bear

you do what you said to make a new quadratic equation. if this has a repeated root then the line must be a tangent to the curve.

so does the repeated root mean that the lines share the same point. Say you were not to get a repeated root, would that just mean the tangent and the curve meet twice?

Reply 4

thekidwhogames

You can look at the intersection equation and prove there's one real root (repeated) by considering discriminant/solving

Reply 5

Appazap

Original post by Tubbz

Yeah you basically prove a point is on both the curve and the line which means they touch at that point, then prove the line and curve have the same gradients at that point.

okay thank you!!!! I dont know whether to give myself marks as my entire answer was:

If the line is a tangent to the curve, they would share the same gradient" then obviously i found x =2.

This is alevel ocr btw

Reply 6

the bear

Original post by Appazap

so does the repeated root mean that the lines share the same point. Say you were not to get a repeated root, would that just mean the tangent and the curve meet twice?

yes; the line would then form a "secant".

if the quadratic had no roots then the line would miss the curve completely.

Reply 7

Svesh

you do 8-2x=x^2-6x+12 and find the solution.
If it is a tangent it should only have one root as it cuts the curve at one point.

Reply 8

Appazap

Original post by aceplayyt

you do 8-2x=x^2-6x+12 and find the solution.
If it is a tangent it should only have one root as it cuts the curve at one point.

hmmn how many marks do you think my working ou got for that question, it was 4 marks?

Reply 9

Svesh

Original post by Appazap

hmmn how many marks do you think my working ou got for that question, it was 4 marks?

Dunno what you meant by -2=2x-6

but I guess upto working out x=2 would be about 3 marks then saying it is a tangent due to there being 1 root would be the 4th mark

How to show a line is the tangent in a curve?

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