TrueTears question thread

[TrueTears]

ah yes yes yes, thank you so much.

[kamil9876]

/0 is right, but analytically speaking we havn't ruled out the possibility that there may be more tangents fiting this information. It's probably the only one as the question said to 'use ur diagram' and other heurestical reasoning about the geometry of the situation sort of leads to this conclusion. A way that this may be done analytically, if ur interested, is to first of all find dy/dx (which i assume u hav already done) and then consider the tangent's gradient. The tangent goes through some point on the hyperbola (x,y) and it satisfies the conditions specified by dy/dx. hence u must now find values that satisfy the equation:



You must also make the substituion using the equation of the hyperbola to get rid of either x or y so that only one variable is involved. This is a fuckload of algebra that is beyond what the question was probably designed for, I think some geometrical reasoning can show that indeed x=3 is the only one. Also, the word "the" in "state the equation of the other tangent to the hyperbola" hints that it is the only one.

[TrueTears]

nice thanks

[TrueTears]

1. In the triangle OAB, let M, N and P be the midpoint of the sides OB, AB and OA respectively. Let the perpendicular bisectors of sides AB and OA meet at G as shown. Let the position vector of P and M be p and m respectively and let be denoted by r as shown.



a) show that m = p + r

this i have done

b) Let be detnoed by v and be denoted by w, as shown. One possible expression for is = v - p + m = v+r, by a)
Use this expression to show that . p = r . p

how do you do b)?

2. A radar tracking facility tracks a plane flying in a straight line. With the radar facility as the origin, the plane's initial and final position vectors are a = 2i + 8j + k and b = 8i -4j + 13k respectively.
a) Find a unit vector , u , in the direction of the motion of the plane.

this i found to be

b) The position vector of the plane, p, when it is closest to the radar can be expressed as p = a + u, where is a real number. find the value of and hence the plane's position when it is closest to the radar facility.

Any help on b)? I've tried letting p = ai +bj +ck but that just gets too many equations and can't be solved.

thanks guys!

[kamil9876]

for b, use the distributive properties of the dot product:

GM.p=p.(v+r)
       =p.v + p.r
but p.v=0 as indicated on the diagram(perpendicular to each other)

[kamil9876]

for the next question, refer to attatched diagram. OA=a, OP=p and OB=b.

OP.AB=0
0=(a+u)(b-a)
0=a.b-a.a+u.b-u.a

because u know a, b, u you can evaluate all those dot products and end up getting some number, with some s in there that you can then work out as they will be the only unkown.

[TrueTears]

thanks kamil for all your help, I got both of them now XD

[TrueTears]

Another question:

Lines y = 2x and y = 10-2x are the asymptotes of a hyperbola of the form.

a) use the equations of the asymptotes to establish a relation between a and b

Just stuck on this part lol

[Mao]

the asymptotes are:



from the given gradients of the asymptotes, (q=5, p=2.5)

[TrueTears]

oh i see lol, stupid me

[TrueTears]

another one

Points A, B and C have position vectors andrespectively.

Find AC : BC

i keep getting 5:-3

but can you have a negative ratio? Or is that wrong lols

[Mao]

AC and BC are scalar quantities, denoting the length of and

so yah, shouldn't be negative.

[TrueTears]

z.z.z. lol its meant to be the magnitude ratio not their vectors hahahaha

[Over9000]

For that big triangle question, how do you do part c
show that GM.m =0

[TrueTears]

so