Complex Numbers
Complex Numbers
star_batch_jee_advanced_2025
Grade 11
Question:
Equation of tangent drawn to the circle $|z|=r$ at the point $A\left(z_0\right)$, is :
Re\left(\frac{z}{z_0}\right)=1
Re\left(\frac{z_0}{z}\right)=1
Im\left(\frac{z}{z_0}\right)=1
Im\left(\frac{z_0}{z}\right)=1
Step-by-Step Solution
Key Concept: The tangent line at any point on a circle is perpendicular to the radius at that point, requiring the perpendicularity condition $\text{Re}(z\overline{z_0})=|z_0|^2$.
For a circle $|z|=r$ with center at origin, the tangent at point $A(z_0)$ where $|z_0|=r$ is perpendicular to the radius $OA$. The radius direction is given by $z_0$, so the tangent must be perpendicular to $z_0$. A point $z$ lies on the tangent if $(z-z_0) \perp z_0$, which means $\text{Re}[(z-z_0)\overline{z_0}]=0$. Expanding: $\text{Re}(z\overline{z_0}) = \text{Re}(z_0\overline{z_0}) = |z_0|^2 = r^2$. Dividing by $|z_0|^2 = r^2$ gives $\text{Re}\left(\frac{z\overline{z_0}}{|z_0|^2}\right)=1$, which simplifies to $\text{Re}\left(\frac{z}{z_0}\right)=1$.
Correct Answer: 1