Parabola

To solve the problem of transforming the equation $y^2=4 a x$ when the coordinate axes are rotated by $45^{\circ}$ anticlockwise about the origin, we start by expressing the new coordinates ( $x, y$ ) in terms of the original coordinates ( $X, Y$ ).Given that the coordinates are rotated $45^{\circ}$ anticlockwise, we can use the following transformation equations:$\;x = X \cos 45^{\circ} - Y \sin \;45^{\circ} = \frac{X}{\sqrt{2}} - \frac{Y}{\sqrt{2}} = \frac{X-Y}{\sqrt{2}}$$\;y = X \sin 45^{\circ} + Y \cos 45^{\circ} = \frac{X}{\sqrt{2}} + \frac{Y}{\sqrt{2}} = \frac{X+Y}{\sqrt{2}}$Substitute these expressions into the given equation $y^2=4 a x$ :$\left( \frac{X+Y}{\sqrt{2}} \right)^2 = 4 a \left( \frac{X-Y}{\sqrt{2}} \right)$Simplifying the equation, we get:$\; \frac{(X+Y)^2}{2} = 4 a \; \frac{X-Y}{\sqrt{2}}$Multiply through by 2 to eliminate the denominator:$(X+Y)^2 = 4 \sqrt{2} a (X-Y)$Thus, the transformed equation is:$(x+y)^2 = 4 \sqrt{2} a (x-y)$This is the equation of the curve after rotating the axes by $45^{\circ}$ anticlockwise.