A point z moves in the complex plane such that arg $\left(\frac{z-2}{z+2}\right)=\frac{\pi }{4},$  then the minimum value of ${\left|z-9\sqrt[]{2}-2i\right|}^2$  is equal to________.

Let zc be the centre of the circle

$\frac{z_c-2}{z_c+2}=i\Rightarrow z_c=2i$

Radius of the circle =  $\sqrt[]{4+4}=2\sqrt[]{2}$

Square off minimum distance AB =  ${\left(7\sqrt[]{2}\right)}^2=98$

<p>Let zc be the centre of the circle</p><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1754477098phpaPaDUq.jpeg" alt="" width="271" height="147" loading="lazy"></picture></div></div><p><span contenteditable="false"> <math> <mrow> <mfrac> <mrow> <msub> <mrow> <mi>z</mi> </mrow> <mrow> <mi>c</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <msub> <mrow> <mi>z</mi> </mrow> <mrow> <mi>c</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mi>i</mi> <mo>⇒</mo> <msub> <mrow> <mi>z</mi> </mrow> <mrow> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>i</mi> </mrow> </math> </span></p><p>Radius of the circle =&nbsp;<span contenteditable="false"> <math> <mrow> <mroot> <mrow> <mn>4</mn> <mo>+</mo> <mn>4</mn> </mrow> <mrow></mrow> </mroot> <mo>=</mo> <mn>2</mn> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> </mrow> </math> </span></p><p>Square off minimum distance AB =&nbsp;<span contenteditable="false"> <math> <mrow> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mn>7</mn> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>9</mn> <mn>8</mn> </mrow> </math> </span></p>

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