Calculate the unknown coordinates of a point $B (x_2,y_2)$ on a line with given distance from a known point $A(x_1,y_1)$

Question

I have a line which represents a cross section. I have the coordinates of on its starting point. I need the coordinates of the end point of that cross section line. The distance between these two points is known. Is there some way to calculate it, or do I need some more information?

I want to calculate the unknown coordinates of end point $B (x_2,y_2)$ on a line with given distance from a known coordinates of starting point $A(x_1,x_2)$.

Clarification

I have a cross section line AB with length of $2800$m. The start point $A(x_1,y_1,z_1)$ is known. $z_1$ is the bed elevation on point $A$. I need to calculate the coordinates at end point $B(x_2,y_2)$. Now there are two situations.

• Case 1: I have $z_2$ at point $B$ with me also.
• Case 2: I don't have $z_2$ on point $B$ with me.

Answer 1

Assuming no typo in what you wrote and that I properly understood (this does not seem to be sure according to the comments I received after my initial answer), if you have point $A$ $(x_1,x_2)$ and point $B$ $(x_2,y_2)$ the square of the distance is given by $$d^2=(x_1-x_2)^2+(x_2-y_2)^2$$

I am sure that you can take from here.

Answer 2

Let $(d)$ be the given line through unknown point $A(x_1, y_1)$. Then the parametric of $d$ has form $$ x=x_0+ta; \quad y=y_0+tb\quad (t\in\mathbb{R}), $$ where $(a, b)$ is the direction vector of $(d)$ and $(x_0, y_0)$ is the point on $(d)$. Let $d$ be the given distance from a known point $B(x_2, y_2)$ to $A$. Solving the equation $$ (x_0+ta-x_2)^2+(y_0+tb-y_2)^2=d. $$ We have three cases:

Case 1. The equation has no solution

Thẹn the distance from known point $B(x_2, y_2)$ to $(d)$ is greater than $d$ and so there is no point $A(x_1, y_1)$ on the line $(d)$ satisfying the given conditions.

Case 2. The equation has one solution

Then substitute the solution $t^*$ to the parametric form of $(d)$ we can find unknown point $A(x_1, y_1)$. The finding point is the projection from known point $B(x_2, y_2)$ to the line $(d)$.

Case 3. The equation has two distinct solutions

Then the distance from known point $B(x_2, y_2)$ to $(d)$ is less than $d$ and there are two point $A_1, A_2$ satissfying the given conditions. Substitute the solutions $t_1, t_2$ to the parametric form of $(d)$ we can find two unknown point $A_1(x^1_1, y^1_1)$ and $A_2(x^2_1, y^2_1)$.