A light ray travelling parallel to the x-axis strikes the outer reflecting surface of a sphere at a point (2, 2, 0) m. Its center is at the point (0, 0, −1) m. The unit vector along the direction of the reflected ray is xî + yĵ + zk̂. Find the value of yz/(4x²).
Physics · Mixed Chapter
JEE Mains

Question

A light ray travelling parallel to the x-axis strikes the outer reflecting surface of a sphere at a point (2, 2, 0) m. Its center is at the point (0, 0, −1) m. The unit vector along the direction of the reflected ray is xî + yĵ + zk̂. Find the value of yz/(4x²).

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: The sphere's centre is C=(0,0,−1) and the incidence point is P=(2,2,0). Outward unit normal n=(P−C)/|P−C|=(2,2,1)/3. Step 2: The incident ray travels in direction d=î. Reflection in the tangent plane gives d′=d−2(d·n)n=(1,0,0)−(4/3)(2,2,1)/3=(1/9,−8/9,−4/9). Step 3: Hence x=1/9, y=−8/9, z=−4/9 and yz/(4x²)=[(32/81)]/[4/81]=8.