JEE 2022 — Mathematics PYQ

To find the number of intersection points, we need to analyze the geometry of the two given equations in the complex plane.

1. Analyze the first equation: $|z|=3$

The equation $|z|=3$ represents a circle centered at the origin $(0,0)$ with a radius of $3$ units.

2. Analyze the second equation: $\arg (z-1)-\arg (z+1)=\frac{\pi }{4}$

This equation can be rewritten using the property $\arg (z_1)-\arg (z_2)=\arg \left(\frac{z_1}{z_2}\right)$:

Geometrically, the expression $\arg (z-z_1)-\arg (z-z_2)=\theta$ represents a circular arc passing through the points $z_1$ and $z_2$.

• In this case, the points are $A(1,0)$ and $B(-1,0)$.

• Since $\theta =\frac{\pi }{4}$ is positive and less than $\pi$, the locus is a major arc of a circle lying in the upper half-plane (because the angle is measured from $z+1$ to $z-1$ counter-clockwise).

3. Find the center and radius of the arc's circle:

Let the circle equation for the arc be $x^2+(y-k{)}^2=r^2$.

• Since it passes through $(1,0)$, we have $1^2+(0-k{)}^2=r^2⟹1+k^2=r^2$.

• The angle subtended by the chord $AB$ (length 2) at the circumference is $\frac{\pi }{4}$. Therefore, the angle subtended at the center is $2\times \frac{\pi }{4}=\frac{\pi }{2}$ (a right angle).

• In the triangle formed by the center $(0,k)$, the origin $(0,0)$, and the point $(1,0)$, the angle at the center is $\frac{\pi }{4}$.

• $\tan (\frac{\pi }{4})=\frac{1}{|k|}⟹1=\frac{1}{|k|}⟹k=1$ (since the arc is in the upper half-plane).

• Radius $r^2=1^2+1^2=2⟹r=\sqrt{2}\approx 1.414$.

4. Compare the two loci:

• Locus 1: A circle centered at $(0,0)$ with radius $R=3$.

• Locus 2: An arc of a circle centered at $(0,1)$ with radius $r=\sqrt{2}$.

Let's check the maximum height reached by the arc. The highest point on the arc is $y=k+r=1+\sqrt{2}\approx 2.414$.

Since the maximum height of the arc ($2.414$) is less than the radius of the first circle ($3$), the arc lies entirely inside the circle $|z|=3$.

Therefore, the circle $|z|=3$ and the arc $\arg (z-1)-\arg (z+1)=\frac{\pi }{4}$ do not intersect at any point.

Final Answer:

The two curves intersect nowhere. The correct option is (C).