Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2014-09-10 04:15:54

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

coordinate

If the vertices of a triangle are

, then find its orthocentre.


friendship is tan 90°.

Offline

#2 2014-09-10 06:20:54

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: coordinate

hi Niharika,

(1) Get an expression for the gradient of one side.

(2) The gradient of the altitude to that side will be minus the reciprocal of that answer.

(3) Hence get the equation for one altitude in the form y = mx + c.

(4) Use the symmetry of the coordinates to write down the equation of a second altitude.

(5)  Subtract to find the x coordinate where they intersect.

You'll find there is some simplification at each step and the final result is quite simple.  smile

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

#3 2014-09-10 06:25:59

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: coordinate

Not sure if this is correct, but I am getting:

Last edited by anonimnystefy (2014-09-10 06:31:09)


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

#4 2014-09-10 06:28:09

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: coordinate

That's what I have.  Please hide your answer so N has a chance to do it before seeing yours.  smile

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

#5 2014-09-10 06:32:18

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: coordinate

Yes, sorry. I was just checking to see if I have messed up.

I need to get into form a bit, haven't done much math in the last 3 months.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

#6 2014-09-10 12:49:22

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: coordinate

Hi bob, I am stuck at step 2


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

#7 2014-09-10 12:54:04

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: coordinate

What are you stuck with?


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

#8 2014-09-10 19:42:36

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: coordinate

I cannot parse the sentence. Can you rephrase it?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

#9 2014-09-10 19:50:51

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: coordinate

hi Agnishom,

If one line has a gradient m, then a perpendicular line will have gradient -1/m.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

#10 2014-09-10 20:51:12

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: coordinate

Yes, that is true.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

#11 2014-09-11 01:17:50

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: coordinate

Did you get the answer?


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

#12 2014-09-11 01:22:19

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

Re: coordinate

are you sure x=a and not -a.
I am getting x=-a.


friendship is tan 90°.

Offline

#13 2014-09-11 01:26:46

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: coordinate

Hm, what are your altitude equation?


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

#14 2014-09-11 03:49:42

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: coordinate

yes, x = -a is good.

And the expression for y is symmetrical in t; by which I mean if you swap over t1 and t2  (or t3) the answer is unchanged.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

#15 2014-09-11 04:11:51

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

Re: coordinate

thanks bob. smile


friendship is tan 90°.

Offline

#16 2014-10-01 19:11:03

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

Re: coordinate

Here is another question:-

Obtain the condition that one of the straight lines given by the equation
                                                         


may coincide with one of those given by the equation
                                                         a'x^2+2h'xy+b'y^2=0


friendship is tan 90°.

Offline

#17 2014-10-01 20:21:03

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: coordinate

hi Niharika,

I'm not at all sure about this at the moment so I'll work on it.  Here's what I know:

In the theory of conic sections, you start with a cone (and an inverted identical cone sitting so their vertices coincide) and then take slices through to generate a circle, ellipse, parabola or hyperbola.  Some sources also recognise a 5th section: if the slice cuts through the edges of the cone then the section is a pair of straight lines.  Some sources call this a degenerate case.

The equation for this is:

Each bracket gives one of the line equations.

As x = y = 0 is a point of your system the equation in this case must simplify to

and if a second system has one line in common, it must have the form

So we end up with:

That's as far as I've got for now.  Hopefully back later.  smile

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

#18 2014-10-02 00:26:20

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: coordinate

Ok.  Some progress.  The above has too many variables for my liking so I thought I'd reduce it by writing the lines in the form y = mx.

As I haven't found a way to do 'a-dash' in Latex, I'll use capitals for a', b' and h' ... ie. A, B and H.

So

This leads to

and similarly

You can eliminate m from these by dividing the first by the third and subtracting the fourth from the second:

From which you can get m, n and p each in terms of a, A, b, B, h and H.

These are very similar expressions, so I suspect  you can do something to show perhaps m^2 = np let's say.  I haven't tried this yet so that might not be it.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

#19 2014-10-02 00:38:53

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: coordinate

Actually, I've just got

I think that will do.  smile

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

#20 2014-10-02 16:16:34

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

Re: coordinate

thanks bob. smile

I didn't consider  fact that y-mx is the factor of both the equations, but yes you used it, so
if we have that (y-mx) is the factor of both the given equations,
why don't we put y=mx in both of them, then eliminate the term x^2 and solve by cross multiplication.

that gives m^2 =(hA-Ha)/(bH-hB)={(aB-Ab)/2(bH-Bh)}^2

That has given a clear solution without m,n and p and the steps are less complicated.

Last edited by niharika_kumar (2014-10-02 16:17:10)


friendship is tan 90°.

Offline

#21 2014-10-02 19:36:43

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: coordinate

hi Niharika,

Your suggestion looks good to me.  Well done for finding a quicker way. 

More good news:  I can now do this

My thanks to Olinguito for this.  smile

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

#22 2014-10-02 22:34:55

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

Re: coordinate

my thanks to him too. smile

Even I was unable to do that and hence my 2nd equation was not in latex form.


friendship is tan 90°.

Offline

#23 2014-10-03 16:23:26

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

Re: coordinate

here is one more:-

Show that all the chords of the curve

which subtend a right angle at the origin pass through a fixed point.Also find that point.


friendship is tan 90°.

Offline

#24 2014-10-04 08:39:49

Bob
Administrator
Registered: 2010-06-20
Posts: 10,058

Re: coordinate

hi Niharika,

I've tried this, on and off, all day and I cannot come up with a neat solution.  The best I can offer has horrible algebra:

(1)  Call the lines y = mx and y = -x/m

(2) Get the coordinates, in terms of m, where these intersect the curve.

(3)  Hence the equation of a chord in terms of m.

(4) Find an 'm' independent point on this line.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

Board footer

Powered by FluxBB