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∘=‌−‌=‌‌y=Xsin‌45∘+Y‌cos‌45∘=‌+‌=‌Substitute these expressions into the given equation
y2=4ax :
(‌)2=4a(‌)Simplifying the equation, we get:
‌=4a‌Multiply 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.