Question

Find the new equation of the conicoid 9x^2 + 16y^2 - 36z^2 - 36x - 72z = 144 when the coordinate system is changed into a new system with the same origin at (-2, 0, 1) and direction ratios same as the old system.

Answer 1

Answer on Question #45098 – Math – Analytic Geometry

Question:

Find the new equation of the conicoid $9x^{2} + 16y^{2} - 36z^{2} - 36x - 72z = 144$ when the coordinate system is changed into a new system with the same origin at $(-2, 0, 1)$ and direction ratios same as the old system.

Solution.

Let denote axes of new coordinate system $u, v, w$.

As a new system is with the origin at $(-2, 0, 1)$ and direction ratios same as the old system, hence we can conclude that $u = x - 2$, $v = y$, $w = z + 1$.

Hence, we have

$$x = u + 2, \quad y = v, \quad z = w - 1$$

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

$$9 (u + 2) ^ {2} + 1 6 v ^ {2} - 3 6 (w - 1) ^ {2} - 3 6 (u + 2) - 7 2 (w - 1) = 1 4 4$$

After simplification, we get

$$9 u ^ {2} + 1 6 v ^ {2} - 3 6 w ^ {2} - 1 4 4 = 0$$

So, the new equation of the conicoid $9x^{2} + 16y^{2} - 36z^{2} - 36x - 72z = 144$ is

$$9 u ^ {2} + 1 6 v ^ {2} - 3 6 w ^ {2} - 1 4 4 = 0$$

Answer. $9u^{2} + 16v^{2} - 36w^{2} - 144 = 0.$

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