bk2 p105 q27
question : the vertices of a quadrilateral are the centres of the circles:
$\displaystyle C_1 : x^2 +y^2+2tx=0$
$\displaystyle C_2: x^2+y^2 +\frac {2y}t = 0$
and their intersecting points
a) find the coordinates of the vertices of the quadrilaterl.
b) find that the area of the quadrilateral is a constant.
my working
vertices :
-t, 0
$\displaystyle 0,- \frac 1 t$
0,0
$\displaystyle -\frac {2t}{1+t^4} , -\frac {2t^3 }{1+t^4}$
area:
$\displaystyle \frac 1 2 \begin{vmatrix} 0 & 0 \\ -t & 0 \\ 0 & -\frac 1 t \\ -\frac {2t}{1+t^4} & -\frac {2t^3 }{1+t^4}\\ 0 & 0 \end{vmatrix}$
$\displaystyle = \frac 1 2 (\frac {t^4-1}{1+t^4})$
cannot prove that the area is a constant
thanks

2. Originally Posted by afeasfaerw23231233
bk2 p105 q27
question : the vertices of a quadrilateral are the centres of the circles:
$\displaystyle C_1 : x^2 +y^2+2tx=0$
$\displaystyle C_2: x^2+y^2 +\frac {2y}t = 0$
and their intersecting points
a) find the coordinates of the vertices of the quadrilaterl.
b) find that the area of the quadrilateral is a constant.
my working
vertices :
-t, 0
$\displaystyle 0,- \frac 1 t$
0,0
$\displaystyle -\frac {2t}{1+t^4} , -\frac {2t^3 }{1+t^4}$
Yes the co-ordinates are right.

But the area is wrong. The area is a constant and it is 1.

I did it the long way. First let $\displaystyle O_1O_2$ be the line joining the centers of $\displaystyle C_1$ and $\displaystyle C_2$.

Then equation of the line $\displaystyle O_1O_2$ is $\displaystyle \frac{x}{t} + yt + 1 = 0$.

Now the base of the triangle is $\displaystyle |O_1O_2| = \sqrt{t^2 + \frac1{t^2}}$.

The perpendicular distance(height) from $\displaystyle (0,0)$ to the line is $\displaystyle \frac{|\frac{0}{t} + 0t + 1|}{\sqrt{t^2 + \frac1{t^2}}}$. So the area of the triangle is $\displaystyle \frac12 |O_1O_2|\frac{|\frac{0}{t} + 0t + 1|}{\sqrt{t^2 + \frac1{t^2}}} = \frac12 |\frac{0}{t} + 0t + 1|= \frac12$.

The perpendicular distance(height) from $\displaystyle (-\frac {2t}{1+t^4} , -\frac {2t^3 }{1+t^4})$ to the line is $\displaystyle \frac{|\frac{-\frac {2t}{1+t^4}}{t} -\frac {2t^3 }{1+t^4}t + 1|}{\sqrt{t^2 + \frac1{t^2}}}$. So the area of the triangle is $\displaystyle \frac12 |O_1O_2|\frac{|-\frac {2}{1+t^4} -\frac {2t^4 }{1+t^4} + 1|}{\sqrt{t^2 + \frac1{t^2}}} = \frac12 |-\frac {2+2t^4 }{1+t^4} + 1| =\frac12 |-2 + 1| = \frac12$.

So the area of the quadrilateral is the sum of the area of triangles.

And that is 1, a constant.

3. Originally Posted by afeasfaerw23231233
bk2 p105 q27
question : the vertices of a quadrilateral are the centres of the circles:
$\displaystyle C_1 : x^2 +y^2+2tx=0$
$\displaystyle C_2: x^2+y^2 +\frac {2y}t = 0$
and their intersecting points
a) find the coordinates of the vertices of the quadrilaterl.
b) find that the area of the quadrilateral is a constant.
my working
vertices :
-t, 0
$\displaystyle 0,- \frac 1 t$
0,0
$\displaystyle -\frac {2t}{1+t^4} , -\frac {2t^3 }{1+t^4}$
area:
$\displaystyle \frac 1 2 \begin{vmatrix} 0 & 0 \\ -t & 0 \\ 0 & -\frac 1 t \\ -\frac {2t}{1+t^4} & -\frac {2t^3 }{1+t^4}\\ 0 & 0 \end{vmatrix}$
$\displaystyle = \frac 1 2 (\frac {t^4-1}{1+t^4})$
cannot prove that the area is a constant
thanks
When t = 1, A = 1. But your answer gives A = 0 ......

4. Originally Posted by mr fantastic
When t = 1, A = 1. But your answer gives A = 0 ......
$\displaystyle A = \frac{1}{2} \, |(x_1 y_2 - x_2 y_1) + (x_2 y_3 - x_3 y_2) + (x_3 y_4 - x_4 y_3) + (x_4 y_1 - x_1 y_4)|$

where the order of the points is such that 1 connects to 2 connects to 3 connects to 4 connects to 1.

So your answer is wrong because you have the order of the points wrong in the formula you used ......

You should get $\displaystyle A = \frac{1}{2} \left( \frac{2 t^4}{1 + t^4} + \frac{2}{1 + t^4}\right) = 1$, as expected from the special case I considered in my earlier reply.

5. Originally Posted by afeasfaerw23231233
bk2 p105 q27
question : the vertices of a quadrilateral are the centres of the circles:
$\displaystyle C_1 : x^2 +y^2+2tx=0$
$\displaystyle C_2: x^2+y^2 +\frac {2y}t = 0$
and their intersecting points
a) ...
b) find that the area of the quadrilateral is a constant.
...
Here is a completely different attempt:

1. Calculate the coordinates of the centers of the 2 circles:

$\displaystyle x^2+2tx+{\color{red}t^2}+y^2 = {\color{red}t^2} ~\iff~ (x+t)^2+y^2 = t^2$

$\displaystyle x^2+y^2 +\frac {2y}t +{\color{red} \left(\frac1t \right)^2}={\color{red} \left(\frac1t \right)^2}~\iff~ x^2+\left(y+\frac1t\right)^2= \left(\frac1t \right)^2$

Then the area of the quadrilateral consists of 2 congruent right triangles. The legs of one of these right triangles have the lengthes(?) $\displaystyle R = t$ and $\displaystyle r = \frac1t$.

Therefore the area of the quadrilateral is:

$\displaystyle A = 2 \cdot \frac12 \cdot t \cdot \frac1t = 1$

6. i joined the points in a wrong order.