# Complex no.

• Feb 13th 2011, 04:06 AM
jacks
Complex no.
If $Z_{1},Z_{2}$ and $Z_{3}$ are Complex no. Representing The Vertices of a Triangle inscribed in $|Z|=2$. The

Altitude Through $Z_{1}$ meets the Circumcircle in $P$, Then The Complex no. Corrosponding To $P$ is =

(Ans in Terms of $Z_{1},Z_{2}$ and $Z_{3}$)
• Feb 13th 2011, 04:58 PM
Sudharaka
Quote:

Originally Posted by jacks
If $Z_{1},Z_{2}$ and $Z_{3}$ are Complex no. Representing The Vertices of a Triangle inscribed in $|Z|=2$. The

Altitude Through $Z_{1}$ meets the Circumcircle in $P$, Then The Complex no. Corrosponding To $P$ is =

(Ans in Terms of $Z_{1},Z_{2}$ and $Z_{3}$)

Dear jacks,

Let, $z_1=x_1+iy_1$

$z_2=x_2+iy_2$

$z_3=x_3+iy_3$

Then write the gradient of the $Z_{2}Z_{3}$ line. Let it be $m_1$.

Since $Z_{2}Z_{3}$ and $PZ_{1}$ are perpendicular you could find the gradient of the $PZ_{1}$ line by,

$m_{1}\times m=-1$

Then you could fine the equation of the $PZ_{1}$ line. Hint: You know the gradient and a point on the $PZ_{1}$ line. Let the equation be $y=mx+c$----(1)

The equation of the circle could be written as, $\mid Z\mid=2\Rightarrow{x^2+y^2=4}$-----------(2)

Using equations (1) and (2) you can find the point P.