Coordinate geometry query

[Inquisitus]

Not actually homework, but just a general query...

I'm trying to write a program that will predict the trajectory of a moving point by analysing several positions through which the point has passed.

Part of this involves finding the coordiantes of a point B, given the coordinates of another point A, the length of AB and its gradient. Obviously there are two possible points... one to the right A and one to the left, but I'm only interested in the one to the right.

I basically need two simple formulae in terms of the x and y coordinates of B, which I can apply in the program itself, although I wouldn't mind knowing how the formula was reached as well

I've tried getting my head around it but I'm just getting confused. Any help would be greatly appreciated, thanks!

 

[StatusX]

I'm not sure what you mean by the gradient. Is that the slope of the line connecting A to B? Are you asking how to find a point B=(x₂,y₂) given a point A=(x₁,y₁), and the length L and slope m of AB?

If so, this can be found by pretending A is at (0,0) (you can add the coordinates of A back in at then end). Then the line containing AB has the equation y=mx, so that points along the line have coordinates like (x,mx). The distance from A to a point (x,mx) is $\sqrt{x^2+m^2 x^2}=x\sqrt{1+m^2}$. Then you want to solve for x using $L= x\sqrt{1+m^2}$. This x is the difference between the x values of A and B, and multiplying it by m gives the difference between the y values.

 

[Architeuthis Dux]

Hi there,

Thank you for reaching out with your query about coordinate geometry. It sounds like you are working on a very interesting program that involves predicting the trajectory of a moving point. In order to find the coordinates of a point B, given the coordinates of another point A, the length of AB, and its gradient, there are a few different formulas that you can use.

One formula that may be helpful is the slope-intercept form of a line, which is y = mx + b. In this formula, m represents the gradient and b represents the y-intercept. You can use this formula to find the coordinates of point B by plugging in the coordinates of point A and the length of AB. From there, you can solve for the y-coordinate of point B and then use the Pythagorean theorem to find the x-coordinate.

Another formula that may be useful is the distance formula, which is d = √((x2-x1)^2 + (y2-y1)^2). In this formula, (x1, y1) represents the coordinates of point A and (x2, y2) represents the coordinates of point B. You can use this formula to find the distance between point A and point B, and then use the given length of AB to solve for the coordinates of point B.

I hope this helps and gives you a starting point for your program. As for how these formulas were derived, they are based on mathematical principles and equations such as the Pythagorean theorem and the slope formula. If you would like more information on the derivation of these formulas, I would recommend doing some research on coordinate geometry or consulting with a mathematics expert.

Best of luck with your program!

Sincerely,