Question

The points $A(a,0),B(0,b),C(c,0)\&D(0,d)$ are such that $ac=bd\&a,b,c,d$ are all non zero. The points

• A. form a parallelogram
• B. do not lie on a circle
• C. form a trapezium
• D. are concyclic

Answer 1

The correct option is D are concyclic

Given the p-oints A(a,o),C(c,o) and (o,d)

such that ac=bd

A quadrilateral is inscribable in a circle if and only if the product of the measure of its diagonals is equal to the sum of the product of the measures of pairs of opposite sides In our care ,it is obvious from mental diagram that diagonals $\left|\overline{(a,o)(c,o)}\right|and\left|\overline{(o,b)(o,d)}\right|\left|\overline{(a,o)(c.o)}\right|.\left|\overline{(o,b)(o,d)}\right|=2ac\sqrt{bd}=2ac\sqrt{ac}=2(ac{)}^{\frac{3}{2}}\left|\overline{(a,o)(o,d)}\right|.\left|\overline{(c,o)(o,b)}\right|+\left|\overline{(o,d)(c,o)}\right|\left|\overline{(a,o)(o,b)}\right|=ac\sqrt{bd}+ac\sqrt{bd}=2ac\sqrt{bd}=2(ac{)}^{\frac{8}{2}}$

$\therefore$ They are concycle