88227 When the coordinate axis are rotated through an angle \(\theta\) in anti clockwise direction, if the transformed equation of \(x^{2}+y^{2}+2 x y+2 x+6 y\) \(+\mathbf{1}=\mathbf{0}\) is \((2+\sqrt{3}) \mathbf{X}^{2}+\mathbf{X Y}+(2-\sqrt{3}) \mathbf{Y}^{2}+\mathbf{a X}\)
\(+\mathbf{b} \mathbf{Y}+\mathbf{2}=\mathbf{0}\), then \(3 \mathbf{a}-\mathbf{b}=\)

88228 Let \(C\) be a curve \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y\) \(+c=0\) in a cartesian plane. By rotating the coordinate axis through an angle \(\frac{\pi}{4}\) in the positive direction, if the transformed equation of \(C\) is \(Y^{2}+X Y-X=0\), then \(\left(h^{2}-a b\right)-2 g f=\)

88229 If \(a \alpha^{2}+b \beta^{2}+c \alpha \beta+d=0\) is the transformed equation of \(4 x^{2}+\sqrt{3} x y+5 y^{2}-4=0\) obtained by using \(\quad \alpha=\frac{\sqrt{3}}{2} x+\frac{y}{2}\) and \(\quad \beta=-\frac{x}{2}+\frac{\sqrt{3}}{2} y\), then \(\mathbf{c}(\mathbf{a}+\mathbf{b}+\mathbf{d})=\)

88231 When the origin is shifted to the point \(\left(\frac{3}{2}, \frac{3}{2}\right)\) by the translation of coordinate axes, then the transformed equation of
\(32 x^{2}+8 x y+32 y^{2}-108 x-108 y+99=0\) is

88227 When the coordinate axis are rotated through an angle \(\theta\) in anti clockwise direction, if the transformed equation of \(x^{2}+y^{2}+2 x y+2 x+6 y\) \(+\mathbf{1}=\mathbf{0}\) is \((2+\sqrt{3}) \mathbf{X}^{2}+\mathbf{X Y}+(2-\sqrt{3}) \mathbf{Y}^{2}+\mathbf{a X}\)
\(+\mathbf{b} \mathbf{Y}+\mathbf{2}=\mathbf{0}\), then \(3 \mathbf{a}-\mathbf{b}=\)

88228 Let \(C\) be a curve \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y\) \(+c=0\) in a cartesian plane. By rotating the coordinate axis through an angle \(\frac{\pi}{4}\) in the positive direction, if the transformed equation of \(C\) is \(Y^{2}+X Y-X=0\), then \(\left(h^{2}-a b\right)-2 g f=\)

88229 If \(a \alpha^{2}+b \beta^{2}+c \alpha \beta+d=0\) is the transformed equation of \(4 x^{2}+\sqrt{3} x y+5 y^{2}-4=0\) obtained by using \(\quad \alpha=\frac{\sqrt{3}}{2} x+\frac{y}{2}\) and \(\quad \beta=-\frac{x}{2}+\frac{\sqrt{3}}{2} y\), then \(\mathbf{c}(\mathbf{a}+\mathbf{b}+\mathbf{d})=\)

88231 When the origin is shifted to the point \(\left(\frac{3}{2}, \frac{3}{2}\right)\) by the translation of coordinate axes, then the transformed equation of
\(32 x^{2}+8 x y+32 y^{2}-108 x-108 y+99=0\) is

88227 When the coordinate axis are rotated through an angle \(\theta\) in anti clockwise direction, if the transformed equation of \(x^{2}+y^{2}+2 x y+2 x+6 y\) \(+\mathbf{1}=\mathbf{0}\) is \((2+\sqrt{3}) \mathbf{X}^{2}+\mathbf{X Y}+(2-\sqrt{3}) \mathbf{Y}^{2}+\mathbf{a X}\)
\(+\mathbf{b} \mathbf{Y}+\mathbf{2}=\mathbf{0}\), then \(3 \mathbf{a}-\mathbf{b}=\)

88228 Let \(C\) be a curve \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y\) \(+c=0\) in a cartesian plane. By rotating the coordinate axis through an angle \(\frac{\pi}{4}\) in the positive direction, if the transformed equation of \(C\) is \(Y^{2}+X Y-X=0\), then \(\left(h^{2}-a b\right)-2 g f=\)

88229 If \(a \alpha^{2}+b \beta^{2}+c \alpha \beta+d=0\) is the transformed equation of \(4 x^{2}+\sqrt{3} x y+5 y^{2}-4=0\) obtained by using \(\quad \alpha=\frac{\sqrt{3}}{2} x+\frac{y}{2}\) and \(\quad \beta=-\frac{x}{2}+\frac{\sqrt{3}}{2} y\), then \(\mathbf{c}(\mathbf{a}+\mathbf{b}+\mathbf{d})=\)

88231 When the origin is shifted to the point \(\left(\frac{3}{2}, \frac{3}{2}\right)\) by the translation of coordinate axes, then the transformed equation of
\(32 x^{2}+8 x y+32 y^{2}-108 x-108 y+99=0\) is

88227 When the coordinate axis are rotated through an angle \(\theta\) in anti clockwise direction, if the transformed equation of \(x^{2}+y^{2}+2 x y+2 x+6 y\) \(+\mathbf{1}=\mathbf{0}\) is \((2+\sqrt{3}) \mathbf{X}^{2}+\mathbf{X Y}+(2-\sqrt{3}) \mathbf{Y}^{2}+\mathbf{a X}\)
\(+\mathbf{b} \mathbf{Y}+\mathbf{2}=\mathbf{0}\), then \(3 \mathbf{a}-\mathbf{b}=\)

88228 Let \(C\) be a curve \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y\) \(+c=0\) in a cartesian plane. By rotating the coordinate axis through an angle \(\frac{\pi}{4}\) in the positive direction, if the transformed equation of \(C\) is \(Y^{2}+X Y-X=0\), then \(\left(h^{2}-a b\right)-2 g f=\)

88229 If \(a \alpha^{2}+b \beta^{2}+c \alpha \beta+d=0\) is the transformed equation of \(4 x^{2}+\sqrt{3} x y+5 y^{2}-4=0\) obtained by using \(\quad \alpha=\frac{\sqrt{3}}{2} x+\frac{y}{2}\) and \(\quad \beta=-\frac{x}{2}+\frac{\sqrt{3}}{2} y\), then \(\mathbf{c}(\mathbf{a}+\mathbf{b}+\mathbf{d})=\)

88231 When the origin is shifted to the point \(\left(\frac{3}{2}, \frac{3}{2}\right)\) by the translation of coordinate axes, then the transformed equation of
\(32 x^{2}+8 x y+32 y^{2}-108 x-108 y+99=0\) is