Question

If the origin is shifted to the point (1,1) and the axes are rotated through an angle 45° about this point, then
the transformed equation of the curve x^2+ 2xy + y^2 - 1 = 0 is​

Answer 1

Transformation of co-ordinates

Given: the origin is shifted to the point (1, 1) and the axes are rotated through an angle 45° about this point

To find: the transformed equation of the curve $x^{2}+2xy+y^{2}-1=0$

SOLUTION:

The formulæ for transformation of axes are

• $x=x'\:cos45^{\circ}-y'\:sin45^{\circ}+1$
• $y=x'\:sin45^{\circ}+y'\:cos45^{\circ}+1$
• $\Rightarrow$
• $\quad x=\frac{1}{\sqrt{2}}(x'-y')+1$
• $\quad y=\frac{1}{\sqrt{2}}(x'+y')+1$

Now, $x^{2}+y^{2}+2xy-1=0$

$\Rightarrow (x+y)^{2}-1$

We substitute the transformation formumæ:

$\quad [\frac{1}{\sqrt{2}}(x'-y'+x'+y')]^{2}-1=0$

$\Rightarrow \frac{1}{2}\:(2x')^{2}-1=0$

$\Rightarrow \frac{1}{2}\times 4x'^{2}-1=0$

$\Rightarrow 2x'^{2}-1=0$

After removing the primes $(x',\:y')$, the general form of the equation becomes

$\quad\quad 2x^{2}-1=0$.

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