#circle + tangent

60 messages · Page 1 of 1 (latest)

hollow goblet
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,rotate

warm mapleBOT
obsidian sandal
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Ok, first make it a quadratic equation by substitution

hollow goblet
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3x+k = 2x^2-3x+8

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??

obsidian sandal
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Ok now collect terms on one side

hollow goblet
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2x^2-6x+8-k=0

obsidian sandal
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How many times does a tangent touch a circle

hollow goblet
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one

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once**

obsidian sandal
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How do we prove a quadratic equation only has one solutoon

hollow goblet
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equal to 0

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wait no

obsidian sandal
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Yes

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So make b^2-4ac = 0 and solve

hollow goblet
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aaah

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what’s that called btw

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discriminate

obsidian sandal
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b^2-4ac is the discriminant

hollow goblet
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aaah

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just remembered that shit

obsidian sandal
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By making the circle equal to the line, were basically moving the circle to be touching the X axis instead of the tangent

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So then we just make an equation with the discriminant and solve

hollow goblet
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which equation do I use for the discriminate then

obsidian sandal
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If there is one solution, b^2-4ac = 0

hollow goblet
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B would be -6?

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a is 2

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and c is 8-k

obsidian sandal
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a = 2, b = -6, c = 8-k

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Yea

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Sub those in

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Then you get a linear equation with one variable

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And get k

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:)

hollow goblet
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28-8k=0

obsidian sandal
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Btw, you could get a similar question at a level which says "show that the line does not touch the curve", or "for which values of k does the line not touch the curve", or smth like that, in which case you just make the discriminant < 0 and solve from there

hollow goblet
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I remember that now

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but i swear there are solutions when it is less than 0

hollow goblet
obsidian sandal
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It takes the square root of the discriminant

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If the discriminant is less than 0, then it needs to take a square root of a negative number

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Which is not possible with real numbers

hollow goblet
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ahh I see my bad

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I was thinking about when it goes to an ‘i’

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complex numbers

obsidian sandal
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Yes

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> 0 means "two real, unique solutions"
= 0 means "one repeated solution"
< 0 means "no real solutions"

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But at gcse imaginary is ingored

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And even at a level

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Imaginary only in fm

hollow goblet
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yeah forgot it is ignored in a level

hollow goblet
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@obsidian sandal

obsidian sandal
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I meant curve

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I was going circles in maths today

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So I got used to solving circle tangent problems

hollow goblet
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ahh alright

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cheers mate