Mathematics - Application of Derivative Question with Solution | TestHub

$y^2-4x-4y+4=0\Rightarrow (y-2{)}^2=4x$, a parabola with vertex $(0,2)$, focus $(1,2)$ and $a=1$. Every member of the family of lines passes through $(1,2)$, the focus, so the chords are focal chords. A focal chord making angle $\theta$ with the axis has length $4a{\csc }^2\theta =\frac{4}{{\sin }^2\theta }$; the perpendicular chord has length $\frac{4}{{\cos }^2\theta }$. Area of quadrilateral $=\frac{1}{2}{d}_1{d}_2=\frac{1}{2}\cdot \frac{16}{{\sin }^2\theta {\cos }^2\theta }=\frac{32}{{\sin }^{2}2\theta }\ge 32$ Minimum area $=32$ (at $\theta =45^{\circ }$ )