Question

show whether a composition of a dilation and a translation can be performed in eith
first, consider ( ( d _ { k } circ t _ { langle a , b
angle } ) ( x , y ) ).
( ( d _ { k } circ t _ { langle a , b
angle } ) ( x , y ) = d _ { k } ( x + a , y + b ) )
( ( d _ { k } circ t _ { langle a , b
angle } ) ( x , y ) = ( k x + k a , k y + k b ) )
now, consider ( ( t _ { langle a , b
angle } circ d _ { k } ) ( x , y ) ).
( ( t _ { langle a , b
angle } circ d _ { k } ) ( x , y ) = )
( t _ { langle a , b
angle } ( k + x , k + y ) )
( t _ { langle a , b
angle } ( x , y ) )
( t _ { langle a , b
angle } ( k x , k y ) )
( t _ { langle a , b
angle } ( k - x , k - y ) )
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Explanation:

Step1: Understand the composition of transformations

The composition \((T_{\langle a,b
angle}\circ D_k)(x,y)\) means first applying the dilation \(D_k\) and then the translation \(T_{\langle a,b
angle}\).
The dilation \(D_k\) transforms a point \((x,y)\) to \((kx,ky)\).

Step2: Apply the translation

After dilation, when we apply the translation \(T_{\langle a,b
angle}\) (which adds \(a\) to the \(x -\)coordinate and \(b\) to the \(y -\)coordinate) to the point \((kx,ky)\) obtained from dilation.

Answer:

\(T_{\langle a,b
angle}(kx,ky)\)