Linear coordinate geometry

**mandarep** · January 20th 2011, 03:43 PM

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.

**Prove It** · January 20th 2011, 05:54 PM

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

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

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

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

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

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

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

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

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

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

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

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

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

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

**mandarep** · January 20th 2011, 06:12 PM

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

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