Parabola

To solve the problem of transforming the equation y2=4ax when the coordinate axes are rotated by 45∘ 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∘ anticlockwise, we can use the following transformation equations:
‌x=X‌cos‌45∘−Ysin‌45∘=‌

X

√2

−‌

Y

√2

=‌

X−Y

√2

‌y=Xsin‌45∘+Y‌cos‌45∘=‌

X

√2

+‌

Y

√2

=‌

X+Y

√2

Substitute these expressions into the given equation y2=4ax :
(‌

X+Y

√2

)2=4a(‌

X−Y

√2

)
Simplifying the equation, we get:
‌

(X+Y)2

2

=4a‌

(X−Y)

√2

Multiply through by 2 to eliminate the denominator:
(X+Y)2=4√2a(X−Y)
Thus, the transformed equation is:
(x+y)2=4√2a(x−y)
This is the equation of the curve after rotating the axes by 45∘ anticlockwise.