Coordinate geometry

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April 1st 2008, 06:53 AM
Coach

Coordinate geometry

Dear forum members,

could you please help me?

Find the values of b for which the straight line $y=-x+b$ and the circle $x^2+y^2-2x-2y-0.25=0$ have no common points.

So the distance from the line to the centre of the circle has to be more than the radius, right?

I solved and obtained $b<2-\frac{3}{2}*\sqrt{2}$

but the sample answer says:

$b<2-\frac{3}{2}*\sqrt{2}$
$b>2+\frac{3}{2}*\sqrt{2}$

Why? Because of the square root?

Thank you in advance!
April 1st 2008, 07:41 AM
wingless

When you get $\sqrt{ {\left ( \frac{b}{2} -1 \right )}^2 + {\left ( \frac{b}{2} -1 \right )}^2 } < \frac{3}{2}$ ,

Notice that $\sqrt{x^2}$ is NOT $x$ , but it's $|x|$ .

$\sqrt{ {\left ( \frac{b}{2} -1 \right )}^2 + {\left ( \frac{b}{2} -1 \right )}^2 } < \frac{3}{2}$

$\sqrt{ 2\cdot{\left ( \frac{b}{2} -1 \right )}^2 } < \frac{3}{2}$

$\sqrt{2}\cdot\sqrt{\cdot{\left ( \frac{b}{2} -1 \right )}^2 } < \frac{3}{2}$

$\sqrt{2} \left | \frac{b}{2} -1 \right | < \frac{3}{2}$

Now solving this gives the correct answer.
April 1st 2008, 07:58 AM
Soroban

Hello, Coach!

Quote:

Find the values of $b$ for which the straight line $y\:=\:-x+b$
and the circle $x^2+y^2-2x-2y-0.25\:=\:0$ have no common points.

So the distance from the line to the centre of the circle
has to be more than the radius, right? . . . . Right!

As you found, the center is $(1,1)$ and the radius is $\frac{3}{2}$

I used this distance formula . . .

The distance from a point $(x_1,y_1)$ to the line $ax + by + c \:=\:0$

. . is given by: . $d \;=\;\frac{|ax_1 + by_1 + c|}{\sqrt{a^2+b^2}}$ . . Note the absolute value.

We have point $(1,1)$ and line $x + y - b \:=\:0$

The distance is: . $d \;=\;\frac{|1 + 1 - b|}{\sqrt{1^2+1^2}}$ . . . which is to be greater than $\frac{3}{2}$

So we have: . $\frac{|2-b|}{\sqrt{2}} \:>\:\frac{3}{2}\quad\Rightarrow\quad |2 - b| \:> \:\frac{3\sqrt{2}}{2}$

And we have two inequalities to solve . . .

. . $2 - b \:>\:\frac{3\sqrt{2}}{2}\quad\Rightarrow\quad \text{-}b \:> \:\text{-}2 + \frac{3\sqrt{2}}{2} \quad\Rightarrow\quad\boxed{ b \:< \:2 - \frac{3\sqrt{2}}{2}}$

. . $2-b \:< \:\text{-}\frac{3\sqrt{2}}{2}\quad\Rightarrow\quad \text{-}b \:< \:\text{-}2 - \frac{3\sqrt{2}}{2} \quad\Rightarrow\quad \boxed{b \:> \:2 + \frac{3\sqrt{2}}{2}}$