A parabolic mirror is silvered at inner surface. The equation of the curve formed by its intersection with X-Y plane is given by `y=(x^(2))/(4)`. A ray travelling in X-Y plane along line `y=x+3` hits the mirror in second quandrant and gets reflected. The unit vector in the direction of reflected ray will be

Options

A. `(1)/sqrt(2)(-hat(i)-hat(j))`
B. `(1)/sqrt(2)(hat(i)+hat(j))`
C. `(-2hat(i)-3hat(j))/(sqrt(13)`
D. `(-(2hat(i)+hat(j)))/(sqrt(5)`

Answer

A

Explanation

Point of intersection of two curves
`y=(x^(2))/(4)=x+3`
`implies x=-2, x=1`
`(dy)/(dx) =(x)/(2)`
i.e., `((dy)/(dx))_(x=-2) =-1`
`tanbeta=` slope of incident ray =1
`tanbeta=1`
`beta=45^(0)`

` tan alpha =|((dy)/(dx))_(x=-2)|=1`
`alpha=45^(0)`
`lfloori=lfloorr=0^(o)`
` The reflected ray makes `45^(o)` with X-axis
Then ` (1)/(sqrt(2))(-hat(i)-hat(j))`