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1. Integration+Rate of change+Differential equation
STILL NEED HELP

A particle P moves along a curve such that at any point on the curve, the rate at which its y-coordinate is changing with respect to its x-coordinate is -2x-4.

When P is at the point X on the curve, the gradient of OX is 4/3 and the distance of P from O is 10 units where O is the origin.

Find the possible coordinates for the point X and the corresponding equations of the curve.

Can anyone help solve this problem??

Last edited by tyre; 05-05-2012 at 20:07.
2. Re: Integration+Rate of change+Differential equation
ok the gradient at the poin is 4/3 and the equation for the gradient is -2x-4.
so 4/3 = -2x -4
4 = -6x - 12
-6x = 16
x = -16/6
so the x coordinate would be (-16/6, y)
you know the distance from O is 10 units, so using pythagoras 10(squared) minus (16/6) squared equals the y coordinate squared.
Thats it i think?
3. Re: Integration+Rate of change+Differential equation
equation of the curve would be the integral of dy/dx so intergrating dy/dx you get;
y= -x^2 -4x +c
then sub in the coordinate you just worked out to find c and bingo!
4. Re: Integration+Rate of change+Differential equation
The question gives you two pieces of information, so let's look at each one.

The first line gives you a clue relating to the gradient of the curve. You should know that the rate at which y is changing with respect to x (or dy/dx) is the gradient of the curve.

The question also tells you that the distance from O to X is 10 units. So if X had coordinates (a,b) you could draw a triangle where a represents the value for the base, b represents the value for the height and the distance OX (10) represents the hypotenuse. The gradient of the line OX also tells you the ratio a:b.

From this you should be able to work out the coordinates of X and the equation of the curve.
5. Re: Integration+Rate of change+Differential equation
In the questions there two possible answers for coordinate of X "Find the possible coordinates for the point X"
6. Re: Integration+Rate of change+Differential equation
The X coordinate according to the answers are (-6,-8) and (6,8)
7. Re: Integration+Rate of change+Differential equation