A ray of light coming from the point `(1, 2)` is reflected at a point `A` on the line `x+y+1=0` and passes through the point `(7, -4)` then the orthocentre of triangle whose first side is `5x-y+13=0, 2^(nd)` sides is x-axis and `3^(rd)` side lies along the reflected ray is :

Options

A. `(-1, 2)`
B. `(-2, -3)`
C. `(-2, -1)`
D. `(-3, -2)`

Answer

D

Explanation

A ray of ligth ………….
By using the property of reflection, we can say that reflected ray passes through the image of point `(-1, 2)` in the line `x+y+1=0`,
`i.e. (x-1)/(1) =(y-2)/(1) = -2((1(1)+1(2)+1)/(1^(2)+1^(2)))`
`implies x-1=y-2= -4`
Point `(-3, -2)`
Equation of reflected ray (which passes through the points `(7, -4)` and `(-3, -2)` is :
`y-(-4)= (-2-(-4))/(-3-7)(x-7)`
`implies y+4= (2)/(-10)(x 7)`
`implies -5y-20= x-7`
`implies x+5y+13 = 0` ....(i)
Now the side of triangle are
`x+5y+13=0`, slope `= -(1)/(5)`
`5x-y+13=0`, slope `= 5`
And x-axis, `y= 0`
Clearly 2 sides are `_|_` therefore it is right angle triangle whose orthocentre is vertex at which right angled is made i.e. point of intersection of `x+5y+13=0` and `5x-y+13=0`
`i.e. (-3,-2)`