(i) Given points are A(a,0) and B(0,a)

x_{1} = a , y_{1}= 0, x_{2} = 0, y_{2} = a

By Distance formula,

d(A,B) = √[(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}]

d(A,B) = √[(0-a)^{2}+(a-0)^{2}]

= √(a^{2}+a^{2})

= √(2a^{2})

= a√2 units.

Hence the distance between the points A and B is a√2 units.

(ii) Given points are P(-6,-3) and Q(-1,9)

x_{1} = -6 , y_{1}= -3, x_{2} = -1, y_{2} = 9

By Distance formula,

d(P,Q) = √[(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}]

d(P,Q) = √[(-1-(-6))^{2}+(9-(-3))^{2}]

= √(5^{2}+12^{2})

= √(25+144)

= √169

= 13

Hence the distance between the points P and Q is 13 units.

(iii) Given points are R(-3a, a) and S(a, -2a)

x_{1} = -3a, y_{1}= a, x_{2} = a, y_{2} = -2a

By Distance formula,

d(R,S) = √[(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}]

d(R,S) = √[(a-(-3a))^{2}+(-2a-a)^{2}]

= √(4a)^{2}+(-3a)^{2})

= √(16a^{2}+9a^{2})

= √(25a^{2})

= 5a

Hence the distance between the points R and S is 5a units.