Paragraph: Let Q be the image of the point P (-2,1,-5) in the plane 3 x-2 y+2 z+1=0 . Question: Consider the following: 1. The coordinates of Q are (4,-3,-1) . 2. PQ is of length more than 8 units. 3. The point (1,-1,-3) is the mid-point of the line segment PQ and lies on the given plane. Which of the above statements are correct?

Step-by-step solution

Let Q (x₁, y₁, z₁ ) be the image of the point P . The direction ratios of P Q are 3,-2,2 . ...(i) The Equation of line P Q is x+2 3 = y-1 -2 = z+5 2 =r Coordinates of any point on the line P Q is 3 r-2,-2 r+ 1 and 2 r-5 . Let Q(3 r-2,-2 r+1,2 r-5) be such a point. Let L be the mid point of P Q, L= ( 3 r 2 -2,1-r, r-5 ) Since L lies on the plane 3 x-2 y+2 z+1=0 So, 3 ( 3 r 2 -2 )-2(1-r)+2(r-5)+1=0 17 2 r-17=0 r=2 So, coordinates of Q are (3 2-2,-2 2+1,2 2-5)=(4,-3,-1) ...(ii) Also the mid point of P Q is L= ( 3 2 2