Question

Find the new equation of the conicoid 2x^2+3y^2+5z^2-xy+z=1 when the coordinate system is transformed into a new system with the origin and with the coordinate axes having direction ratios 2,1,0; -1,2,5; 1,-2,1 with respect to the old system.

Answer 1

Answer on Question #45078 – Math – Analytic Geometry

Question:

Find the new equation of the conicoid $2x^{2} + 3y^{2} + 5z^{2} - xy + z = 1$ when the coordinate system is transformed into a new system with the origin and with the coordinate axes having direction ratios 2,1,0; -1,2,5; 1,-2,1 with respect to the old system.

Solution.

Let denote axes of new coordinate system $u, v, w$. As a new system is with the origin and with the coordinate axes having direction ratios 2,1,0; -1,2,5; 1,-2,1 with respect to the old system, hence we can conclude that $u = 2x + y$, $v = x + 2y + 5z$, $w = x - 2y + z$.

Hence, the transformation matrix is

$$\left( \begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & -2 \\ 0 & 5 & 1 \end{array} \right).$$

So, the matrix of inverse transformation is

$$\left( \begin{array}{ccc} 3/7 & \frac{1}{7} & -\frac{1}{7} \\ -\frac{1}{28} & \frac{1}{14} & \frac{5}{28} \\ \frac{5}{28} & -\frac{5}{14} & \frac{3}{28} \end{array} \right).$$

Thus, $x = \frac{3}{7u} - \frac{1}{28}v + \frac{5}{28}w$, $y = \frac{1}{7}u + \frac{1}{14}v - \frac{5}{14}w$ and $z = -\frac{1}{7}u + \frac{5}{28}v + \frac{3}{28}w$.

Substituting this into the equation of the conicoid in the old system, we get

$$2x^{2} + 3y^{2} + 5z^{2} - xy + z = 1$$ $$\begin{array}{l} 2\left(\frac{3}{7} u - \frac{1}{28} v + \frac{5}{28} w\right)^{2} + 3\left(\frac{1}{7} u + \frac{1}{14} v - \frac{5}{14} w\right)^{2} + 5\left(-\frac{1}{7} u + \frac{5}{28} v + \frac{3}{28} w\right)^{2} \\ \quad - \left(\frac{3}{7} u - \frac{1}{28} v + \frac{5}{28} w\right)\left(\frac{1}{7} u + \frac{1}{14} v - \frac{5}{14} w\right) - \frac{1}{7} u + \frac{5}{28} v + \frac{3}{28} w = 1. \end{array}$$

After simplification, we get

$$\frac{23 u^{2}}{49} + \frac{141 v^{2}}{784} + \frac{445 w^{2}}{784} - \frac{55 u v}{196} - \frac{5 u w}{196} - \frac{5 v w}{392} - \frac{u}{7} + \frac{5 v}{28} + \frac{3 w}{28} = 1$$

Answer. The new equation of the conicoid is

$$\frac{23 u^{2}}{49} + \frac{141 v^{2}}{784} + \frac{445 w^{2}}{784} - \frac{55 u v}{196} - \frac{5 u w}{196} - \frac{5 v w}{392} - \frac{u}{7} + \frac{5 v}{28} + \frac{3 w}{28} = 1$$

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