Let T be the line passing through the points P(- 2, 7) and Q (2, -...

$AP = AQ = AM$ So $∠ PMQ = 90 °$
Locus of $M$ is a circle having $PQ$ as its diameter
Hence, ${E}_{1}: {x}^{2} + {y}^{2} - 2y - 39 = 0, x \ne \pm 2$
Clearly \left(- 2, 7\right)$is not included in the locus because we can not consider point circles, and \left(0, \dfrac{3}{2}\right)$ does not lie on ${E}_{1}$ as it does not satisfy the equation above.

The equation of chord with $B\left(h, k\right) as mid point will be given by$T = {S}_{1}$or</p><p>$xh + yk - y - k - 39 = {h}^{2} + {k}^{2} - 2k - 39$</p><p>or$xh + yk - y + k = {h}^{2} + {k}^{2}$</p><p>It passes through$R\left(1, 1\right)$. So,</p><p>$h + k - 1 + k = {h}^{2} + {k}^{2}$</p><p>or the locus of${E}_{2}$is${x}^{2} + {y}^{2} - x - 2y + 1 = 0$</p><p>But if$P$is not the part of locus of$M$then$PQ$is not the chord of locus of$M$. So point \left(\dfrac{4}{5}, \dfrac{7}{5}\right)$ does not lie in ${E}_{2}$. \left(\dfrac{1}{2}, 1\right)$does not satisfy the equation hence does not lie in${E}_{2}.