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Standard X

Mathematics

Intercept of a Straight Line

When the axes...

Question

When the axes are rotated through an angle $45^{\circ},$ the transformed equation of a curve is $17 x^{2} - 16 x y + 17 y^{2} = 225.$ Find the original equation of the curve.

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Solution

Given equation is $17 X^{2} - 16 X Y + 17 Y^{2} = 225$ ........eq(i)
We know that
$X = x c o s θ + y s i n θ$
$Y = - x s i n θ + y c o s θ$
Given $θ = 45^{\circ}$
$\Rightarrow X = \frac{1}{\sqrt{2}} (x + y)$ and $Y = \frac{1}{\sqrt{2}} (- x + y)$
substitute the values X and Y in the given eq(i) we get the following results
So, the transformed equation is
$\frac{17}{2} (x + y)^{2} - 8 (x + y) (- x + y) + \frac{17}{2} (- x + y)^{2} = 225$

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Similar questions

Q. lf the transformed equation of a curve is

17 X^{2} - 16 X Y + 17 Y^{2} = 225

when the axes are rotated through an angle

45^{o}

, then the original equation of the curve is

Q. Transform to axes inclined at

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to the original axes the equations
(1)

x^{2} - y^{2} = a^{2}

,
(2)

17 x^{2} - 16 x y + 17 y^{2} = 225

, and

(3)

y^{4} + x^{4} + 6 x^{2} y^{2} = 2

Q. The equation of a curve is

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. Its equation if the axes are rotated through an angle

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will be .

"WHEN THE ORIGIN IS SHIFTED TO THE POINT (2,3) THE TRANSFORMED EQUATION OF THE CURVE IS Xsquare+3xy-2y square+17x-16xy+17y square=225 find the qriginal equatio of the curve"

Q. Axes are rotated through

a + i v e

obtuse angle

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so that the transformed equation of the curve

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is free from the term of

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then the coefficient of

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When the axes are rotated through an angle 45∘, the transformed equation of a curve is 17x2−16xy+17y2=225. Find the original equation of the curve.

Given equation is 17X2−16XY+17Y2=225 ........eq(i)We know thatX=xcosθ+ysinθY=−xsinθ+ycosθGiven θ=45∘⇒X=1√2(x+y) and Y=1√2(−x+y)substitute the values X and Y in the given eq(i) we get the following resultsSo, the transformed equation is172(x+y)2−8(x+y)(−x+y)+172(−x+y)2=225

When the axes are rotated through an angle $45^{\circ},$ the transformed equation of a curve is $17 x^{2} - 16 x y + 17 y^{2} = 225.$ Find the original equation of the curve.