CUET PG 2023 — Mathematics PYQ

1. Positions relative to the Circle $C$

The equation of the circle is $S_1:x^2+y^2-9=0$.

• For Point $P(1,2)$:

  $$S_1(P)=1^2+2^2-9=1+4-9=-4$$

  Since -4 < 0, $P$ lies inside the circle $C$.

• For Point $Q(2,1)$:

  $$S_1(Q)=2^2+1^2-9=4+1-9=-4$$

  Since -4 < 0, $Q$ lies inside the circle $C$.

2. Positions relative to the Ellipse $E$

The equation of the ellipse is $S_2:\frac{x^2}{9}+\frac{y^2}{4}-1=0$.

• For Point $P(1,2)$:

  $$S_2(P)=\frac{1^2}{9}+\frac{2^2}{4}-1=\frac{1}{9}+\frac{4}{4}-1$$
  $$S_2(P)=\frac{1}{9}+1-1=\frac{1}{9}$$

  Since \frac{1}{9} > 0, $P$ lies outside the ellipse $E$.

• For Point $Q(2,1)$:

  $$S_2(Q)=\frac{2^2}{9}+\frac{1^2}{4}-1=\frac{4}{9}+\frac{1}{4}-1$$

  To compare, find a common denominator:

  $$S_2(Q)=\frac{16+9}{36}-1=\frac{25}{36}-1=-\frac{11}{36}$$

  Since -\frac{11}{36} < 0, $Q$ lies inside the ellipse $E$.

  Final Conclusion:

  • $P$ lies inside $C$ but outside $E$.