Question

$A(p,0),B(4,0),C(5,6)$ and $D(1,4)$ are the vertices of a quadrilateral $ABCD$. If $\angle ADC$ is obtuse, the maximum integral value of $p$ is :

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is B $2$

$\cos \angle ADC=\frac{(AD{)}^2+(CD{)}^2-(AC{)}^2}{2AD.CD}<0$
$\Rightarrow (p-1{)}^2+4^2+(5-1{)}^2+(6-4{)}^2<(p-5{)}^2+(6{)}^2$
$\Rightarrow 8p<24\Rightarrow p<3$ and the maximum integral value of p is $2.$