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Parametric Equations help

hey i need a hand with this Parametric Equations question
C has parametric equations x=1+4t/1-t , y=2+bt/1-t , -1≤t≤ 0
a)Show that the Cartesian equation of C is y=(2+b/5)x+(8-b/5),over an appropriate domain.
Given that C is a line segment and that the gradient of the line is −1,
b)show that the length of the line segment is a√2 , where a is a rational number to be found.

I've not really got any clue where to start thanks in advance.
(edited 5 years ago)
Original post by bhyvuoogyyu
hey i need a hand with this Parametric Equations question
C has parametric equations x=1+4t/1-t , y=2+bt/1-t , -1≤t≤ 0
a)Show that the Cartesian equation of C is y=(2+b/5)x+(8-b/5),over an appropriate domain.
Given that C is a line segment and that the gradient of the line is −1,
b)show that the length of the line segment is a√2 , where a is a rational number to be found.

I've not really got any clue where to start thanks in advance.


For the first part, you want to elimiate the parameter tt somehow. Notice that you can rearrange the first equation to:

x−14=t1−t\dfrac{x-1}{4} = \dfrac{t}{1-t}

and the second to

y−2b=t1−t\dfrac{y-2}{b} = \dfrac{t}{1-t}

So you can substitute the second equation now into the first and eliminate tt.
Original post by bhyvuoogyyu
hey i need a hand with this Parametric Equations question
C has parametric equations x=1+4t/1-t , y=2+bt/1-t , -1≤t≤ 0
a)Show that the Cartesian equation of C is y=(2+b/5)x+(8-b/5),over an appropriate domain.
Given that C is a line segment and that the gradient of the line is −1,
b)show that the length of the line segment is a√2 , where a is a rational number to be found.

I've not really got any clue where to start thanks in advance.


Can you use brackets, please, to show us the equations as accurately as possible?

RDK thinks you mean
x = 1 + 4t/(1-t)

but I have an inkling that you intended
x = (1+4t) / (1-t)

although you have actually written
x = 1 +(4t/1) - t

... and similarly for the y-equations.

Whichever it might be, RDK's method is correct.
Cartesian Eqn means ONLY in terms of x and y, so eliminate t from the equations by substitution. (this is similar to similtaneous eqn processes)

If you manage this, you should get

y = (2+b)x / 5 + (8-b) / 5

Then you need to consider only the small part of this straight line as determined by the allowed values for t.
Think about how the limited values for t will limit the values for x and y/
[this is the hardest part of the question]

Once you have those limits, you'll easily see / calculate the length of the line segment.

Let us know how you get on.
Can someone please help me out in part b of the question above, I’m struggling with it too...
as t changes from -1 to 0, that determines the set of possible values for x and also for y

sub these in to the cartesian equation and you can see that we've got a line segment (part of a straight line)

so you can then calculate the length of that line segment
Original post by begbie68
as t changes from -1 to 0, that determines the set of possible values for x and also for y

sub these in to the cartesian equation and you can see that we've got a line segment (part of a straight line)

so you can then calculate the length of that line segment

Got it! Thank you
I am substituting equation 1 into 2 so it is (x-1)/4 = (y-2)/b but after rearranging i cant get to the answer
Original post by Carboxylic
I am substituting equation 1 into 2 so it is (x-1)/4 = (y-2)/b but after rearranging i cant get to the answer


That's incorrect. Have a look at post #3. The orginal question is poorly ( and incorrectly) bracketed, so that's not the correct equation.

Post should have said: x=(1+4t)/(1-t) , y=(2+bt)/(1-t) and have a go from there.
can you show the working out because its vey confusing
Original post by genius7277
can you show the working out because its vey confusing


Fully worked solutions are against the forum rules, and I await a response from @Carboxylic first of all.
I'm stuck on the same question please could you give me some direction as to what i should do
Original post by jessiepip
I'm stuck on the same question please could you give me some direction as to what i should do


I suggest you read the thread fully to start, have a go, and post working if you're still stuck.

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