All the points on the ellipse (x^2)/(2)+(y^2)/(1)=1 are rotated about origin through an angle (π

step 1 Lips centered at negative 4, negative 2. And I've got a major axis horizontally. That is 6 and a

All the points on the ellipse (x^2)/(2)+(y^2)/(1)=1 are...

All the points on the ellipse $\frac{\mathrm{x}^{2}}{2}+\frac{\mathrm{y}^{2}}{1}=1$ are rotated about origin through an angle $\frac{\pi}{4}$ in counter clockwise sense. The equation of ellipse changes to (a) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}+2 \mathrm{XY}=4$ (b) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}+\mathrm{X} Y=2$ (c) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}-2 \mathrm{XY}=4$ (d) $\frac{X^{2}}{\sqrt{2}}+\frac{Y^{2}}{\sqrt{2}}-X Y=1$

All the points on the ellipse $\frac{\mathrm{x}^{2}}{2}+\frac{\mathrm{y}^{2}}{1}=1$ are rotated about origin through an angle $\frac{\pi}{4}$ in counter clockwise sense. The equation of ellipse changes to
(a) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}+2 \mathrm{XY}=4$
(b) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}+\mathrm{X} Y=2$
(c) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}-2 \mathrm{XY}=4$
(d) $\frac{X^{2}}{\sqrt{2}}+\frac{Y^{2}}{\sqrt{2}}-X Y=1$

Key Concepts

Rotation of Axes

Rotation of axes is a technique used to transform the coordinate system by rotating it through a specified angle. This transformation is performed using formulas that relate old and new coordinates and is particularly useful for eliminating the xy-term in conic sections to simplify the equation and analysis of the curve.

Coordinate Transformation

A coordinate transformation involves changing variables from one coordinate framework to another, such as when the coordinate system is rotated or translated. This method is widely applied to convert geometric equations into forms that are easier to work with or that reveal important properties of the shapes, such as symmetry.

Conic Sections and Ellipses

Conic sections are curves obtained by intersecting a plane with a double-napped cone, with ellipses being one among them. An ellipse is defined by its standard algebraic form, and understanding its properties—including how its equation changes under coordinate transformations—is fundamental in geometry and algebra.

Transformation of Quadratic Forms

Quadratic forms represent equations involving second-degree polynomials in two variables. Transforming a quadratic form, such as by rotating the axis, changes the coefficients and can simplify the representation of conic sections by eliminating the cross-product term, thus revealing the underlying geometry of the curve.

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