Question

(6) Consider the 3-D conic section 5x2 + y2 + 222-412-y+1+2-1-0. Rotate and translate the coordinate axes to write it in standard form. Hence determine the type of surface this describes

Answer 1

Solutlon: Glven 3-D conlc sectlon Is A quadratic surface Is the graph of a second-degree equatlon In three varlables x,y and z; Translatlon:Let (p, q, r) denote the coordlnates of O In the XYZ-system. Conslder now the new system of coordinate axes O'x', O'Y', o'Z parallel to the glven system with orlgln O'. We know that the relatlon between the coordInates In the orlglnal and new system are glven by az Substltuting these values In (1I), we have Now, we expand the above expresslon and slmplify by arranglng like terms. We obtaln, where, (vi) Now, thls conlcold has a cener at the orlgln. Therefore, we have u' vw0. Thls means that

u(ap+ ha+gr) +u 0 w, = (gp + fe+cr) + w Let G (x, y', z') denote the expresslon In the left hand slde of equatlon (v). Then we see that any point (Z' . У, 2, satisfies the equation G (x, y, z)-0, which Is agaln a general second degree equatlon. under the transformatlon of shlfting the orlgln of a coordlnate system, the second degree part of a general second degree equatlon does not change Rotatlon of axes Let, P be any polnt In space having coordlnates (x, y, z) and (x, y', z) wlth respect to the old and new coordlnate systems respectlvely. Let PN be the perpendlcular from P on OY. Then ON-y. The line segnleut ON Is also the projection of OP on the Ⅱne OY' with direction cosines 2, m2, n2. Therefore, we have Therefore, we get slmllarly, we have Therefore glven (x, y, z) and the dlrectlon coslnes of the new co ordlnate axes, we can obtaln the new coordlnates (x, y', z') uslng equatlons(), **and (***) SImllarly, also we can obtaln and

Therefore, transformatlon of coordlnates are glven by 7l m1 12 7l m2 773 m3