# Thread: Linear coordinate geometry

1. ## Linear coordinate geometry

Hopefully this is the right section to be posting this...
The point (h, k) lies on the the line y=x+1 and is 5 units from the point (0, 2). Write down two equations connecting h and k and hence find the possible values of h and k.
Thanks in advance to anyone who can help.

2. Since $\displaystyle \displaystyle y = x + 1$ and $\displaystyle \displaystyle (h, k)$ lies on the line, that means $\displaystyle \displaystyle k = h + 1$.

The distance from $\displaystyle \displaystyle (h, h + 1)$ to $\displaystyle \displaystyle (0, 2)$ is $\displaystyle \displaystyle 5$.

So $\displaystyle \displaystyle \sqrt{(h - 0)^2 + (h + 1 - 2)^2} = 5$

$\displaystyle \displaystyle \sqrt{h^2 + (h - 1)^2} = 5$

$\displaystyle \displaystyle \sqrt{h^2 + h^2 - 2h + 1} = 5$

$\displaystyle \displaystyle \sqrt{2h^2 - 2h + 1} = 5$

$\displaystyle \displaystyle 2h^2 - 2h + 1 = 25$

$\displaystyle \displaystyle 2h^2 - 2h - 24 = 0$

$\displaystyle \displaystyle 2(h^2 - h - 12) = 0$

$\displaystyle \displaystyle h^2 - h - 12 = 0$

$\displaystyle \displaystyle (h - 4)(h + 3) = 0$

$\displaystyle \displaystyle h - 4 = 0$ or $\displaystyle \displaystyle h + 3 = 0$

$\displaystyle \displaystyle h = 4$ or $\displaystyle \displaystyle h = -3$.

Since $\displaystyle \displaystyle k = h + 1$, that means the two possible values for $\displaystyle \displaystyle (h, k)$ are $\displaystyle \displaystyle (-3, -2)$ and $\displaystyle \displaystyle (4, 5)$.

3. Wow! You are brilliant. That makes perfect sense to me. Thanks so much.