Maths

The line with equation mx – y – 2 = 0 touches the circle with equation x 2 + 6x +y 2 – 8y = 4.
Find the two possible values of m, giving your answersin exact form.

This person got this anwser,
y = mx - 2

So we want to find m such that x2 + 6x + (mx - 2)2 - 8(mx - 2) = 4 has a single solution.

x2 + 6x + m2x2 - 4mx + 4 - 8mx + 16 = 4
(m2+1)x2 + (-12m+6)x + 16 = 0

So we need the discriminant to equal 0.

(-12m+6)2 - 64(m2+1) = 0
144m2 - 144m + 36 - 64m2 - 64 = 0
80m2 - 144m - 28 = 0

Now use quadratic formula to find your values for m.

I am confused where the x went? can someone explain in depth please and thanks in advance.

Reply 1

username5340890

9

Do you understand what the discriminant is? If not that’s probably the source of your confusion

Reply 2

HashMash

OP

11

Original post by Flibbler

Do you understand what the discriminant is? If not that’s probably the source of your confusion

Yes I know what a discriminant is. Am i missing something? could you explain

Reply 3

username5340890

9

When you use the discriminant you are certainly going to lose the x. That’s the whole point of the discriminant. You plug in the constants a,b,c from ax^2 + bc +c =0. It’s basically just a part of the quadratic formula.

Since your question is about losing the x, the discriminant might be something to look at.

If you are confused about a different part of the question, let us know

Reply 4

Sinnoh

22

You seem to be following the steps a bit wrong. The x went away because when you take the discriminant, you're not rewriting the earlier equation in terms of x in a different form - you're looking at an entirely different equation that just so happens to also equal 0.

Reply 5

HashMash

OP

11

Original post by Flibbler

When you use the discriminant you are certainly going to lose the x. That’s the whole point of the discriminant. You plug in the constants a,b,c from ax^2 + bc +c =0. It’s basically just a part of the quadratic formula.

Since your question is about losing the x, the discriminant might be something to look at.

If you are confused about a different part of the question, let us know

Oh I see what you mean now. Okay ill give discriminant a better look.

Another question, this may be a bit stupid.
If there were 2 brackets. Lets say
(2x-2)+(5x+4)
would these 2 multiply into each other? Considering there is an addition sign between them.

Reply 6

HashMash

OP

11

Original post by Sinnoh

You seem to be following the steps a bit wrong. The x went away because when you take the discriminant, you're not rewriting the earlier equation in terms of x in a different form - you're looking at an entirely different equation that just so happens to also equal 0.

They are the same equations though aren't they.

Reply 7

Sinnoh

22

Original post by HashMash

They are the same equations though aren't they.

The equation to solve for

x

is

(m^2+1)x^2 + (-12m+6)x + 16 = 0

. You can't solve this (to get specific values) without knowing

m

. You solve for

m

by taking the discriminant to equal 0, which gives you a different quadratic equation. You seemed to suggest in your original post that the discriminant equation was a way of rewriting the earlier equation, which it isn't.

(edited 1 year ago)

Reply 8

HashMash

OP

11

Original post by Sinnoh

The equation to solve for

x

is

(m^2+1)x^2 + (-12m+6)x + 16 = 0

. You can't solve this (to get specific values) without knowing

m

. You solve for

m

by taking the discriminant to equal 0, which gives you a different quadratic equation. You seemed to suggest in your original post that the discriminant equation was a way of rewriting the earlier equation, which it isn't.

Ah and that's when the

x

are taken out. I see thank you.

Reply 9

username5340890

9

Original post by HashMash

Oh I see what you mean now. Okay ill give discriminant a better look.

Another question, this may be a bit stupid.
If there were 2 brackets. Lets say
(2x-2)+(5x+4)
would these 2 multiply into each other? Considering there is an addition sign between them.