All the points on the ellipse of question no.10, after...

All the points on the ellipse of question no.10, after rotation are then shifted two units along positive $\mathrm{x}$ -direction and two units along positive y-direction. The equation then becomes (a) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}+2 \mathrm{X}+2 \mathrm{Y}+2 \mathrm{XY}=4$ (b) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}+2 \mathrm{X}+2 \mathrm{Y}+\mathrm{XY}=2$ (c) $\frac{\mathrm{X}^{2}}{\sqrt{2}}+\frac{\mathrm{Y}^{2}}{\sqrt{2}}-2 \mathrm{X}-2 \mathrm{Y}-\mathrm{XY}=1$ (d) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}-8 \mathrm{X}-8 \mathrm{Y}-2 \mathrm{XY}+12=0$

All the points on the ellipse of question no.10, after rotation are then shifted two units along positive $\mathrm{x}$ -direction and two units along positive y-direction. The equation then becomes
(a) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}+2 \mathrm{X}+2 \mathrm{Y}+2 \mathrm{XY}=4$
(b) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}+2 \mathrm{X}+2 \mathrm{Y}+\mathrm{XY}=2$
(c) $\frac{\mathrm{X}^{2}}{\sqrt{2}}+\frac{\mathrm{Y}^{2}}{\sqrt{2}}-2 \mathrm{X}-2 \mathrm{Y}-\mathrm{XY}=1$
(d) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}-8 \mathrm{X}-8 \mathrm{Y}-2 \mathrm{XY}+12=0$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. Ellipses are one type of conic, characterized by their oval shape, and their equations in the Cartesian plane often require transformation techniques to analyze their properties.

Rotation of Axes

Rotation of axes is a coordinate transformation that reorients the coordinate system by a specific angle. This technique is often used to eliminate the xy (or cross-product) term in the second-degree equation of a conic section, simplifying the analysis and classification of the curve.

Translation of Axes

Translation of axes entails shifting the coordinate system by adding constant values to the x and y coordinates. This transformation is used to reposition the geometric figure, often to move its center to the origin, which simplifies the equation of the conic by removing linear terms.

Coordinate Transformations

Coordinate transformations, including both rotations and translations, are essential in the study of conic sections. They allow one to modify the standard form of an equation to a more workable or recognizable form, facilitating easier interpretation of the geometric properties of the curve.

Transcript

Please give Ace some feedback