QJEEM22Mab642: JEE Main 2022 Mathematics - Class 11 JEEM22Mab642 Solution: A point moves so that the sum of squares of its distances from…

Let point $P:(h,\,k)$

${(h - 1)^2} + {(k - 2)^2} + {(h + 2)^2} + {(k - 1)^2} = 14$

$2{h^2} + 2{k^2} + 2h - 6k - 4 = 0$

Locus of $P:{x^2} + {y^2} + x - 3y - 2 = 0$

Intersection with x-axis,

${x^2} + x - 2 = 0$

$\Rightarrow x = - 2,\,1$

Intersection with y-axis,

${y^2} - 3y - 2 = 0$

$\Rightarrow y = {{3\, \pm \,\sqrt {17} } \over 2}$

Area of the quadrilateral ACBD is

$= {1 \over 2}(|{x_1}| + |{x_2}|)(|{y_1}| + |{y_2}|)$

$= {1 \over 2} \times 3 \times \sqrt {17} = {{3\sqrt {17} } \over 2}$