If \(A (2a+3b, a +6b)\) and \(B (3a + 4b, 2a + 5b)\) are two points, then distance between the points A and B is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\sqrt{2\left(a^{2}+b^{2}\right)}\)
The distance formula gives the distance between \((x_1,y_1)\) and \((x_2,y_2)\) as \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\); it is Pythagoras’ theorem with the two differences as the sides of a right triangle.
\(x_2-x_1=(3a+4b)-(2a+3b)=a+b\)
\(y_2-y_1=(2a+5b)-(a+6b)=a-b\)
\(AB^2=(a+b)^2+(a-b)^2=2a^2+2b^2=2\left(a^{2}+b^{2}\right)\), since the \(2ab\) terms cancel
\(AB=\sqrt{2\left(a^{2}+b^{2}\right)}\)