JEE Advanced 2018 · Paper 2, Mathematics question 2

Locus $E_1$. Let the common tangent at $M$ meet $T$ at $O'$. Tangent lengths from $O'$ give $O'P=O'M=O'Q$. So $O'$ is the midpoint of $PQ$, and $M$ lies on the circle with diameter $PQ$:
$$x^2+(y-1)^2=40,$$
with $P$ and $Q$ themselves excluded (no pair of circles touches there).

• (A) $(-2,7)=P$ is excluded. False.
• (D) $(0,\tfrac32)$: $0+\tfrac14\ne40$, so it is not on $E_1$. True.

Locus $E_2$. The mid-point $N$ of a chord through $R(1,1)$ of the circle centred at $C(0,1)$ satisfies $CN\perp NR$:
$$x(x-1)+(y-1)^2=0.$$
But the chords through $R$ that end at $P$ or $Q$ are not allowed.

• (B) $\left(\tfrac45,\tfrac75\right)$ satisfies the equation: $\tfrac{16}{25}-\tfrac45+\tfrac4{25}=0$. It is the foot of the perpendicular from $C$ to line $RP$ (direction $(-1,2)$; $t=\tfrac15$ gives $(\tfrac45,\tfrac75)$). So it is the mid-point of the chord through $P$, which is excluded. It does NOT lie in $E_2$. True.
• (C) $\left(\tfrac12,1\right)$: $\tfrac14-\tfrac12+0\ne0$. False.

(B), (D)