grandes-ecoles 2011 QIV.A.3

grandes-ecoles · France · centrale-maths2__psi Not Maths

We consider four distinct points $A, B, C$ and $D$ in the canonical Euclidean space $\mathbb{R}^3$ such that $AB = BC = CD = DA = 1$, $AC = a > 0$ and $BD = b > 0$.
Show that if strictly positive reals $a$ and $b$ satisfy the relation $a^2 + b^2 \leqslant 4$, then there indeed exist four distinct points $A, B, C$ and $D$ in the canonical Euclidean space $\mathbb{R}^3$ satisfying $AB = BC = CD = DA = 1$, $AC = a$ and $BD = b$.

We consider four distinct points $A, B, C$ and $D$ in the canonical Euclidean space $\mathbb{R}^3$ such that $AB = BC = CD = DA = 1$, $AC = a > 0$ and $BD = b > 0$.

Show that if strictly positive reals $a$ and $b$ satisfy the relation $a^2 + b^2 \leqslant 4$, then there indeed exist four distinct points $A, B, C$ and $D$ in the canonical Euclidean space $\mathbb{R}^3$ satisfying $AB = BC = CD = DA = 1$, $AC = a$ and $BD = b$.

Paper Questions

QI.A QI.B QI.C QII.A QII.B QIII.A QIII.B QIV.A.1 QIV.A.2 QIV.A.3 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.B.5 QIV.B.6 QV.A.1 QV.A.2 QV.B.1 QV.B.2 QV.B.3 QV.C.1 QV.C.2 QV.C.3 QV.C.4 QV.C.5 QV.C.6 QV.C.7