Let P be the image of the point Q(7,−2,5) in the line L:2x−1=3y+1=4z and R(5, p, q) be a point on L. Then the square of the area of △PQR is:
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Step-by-Step Solution
Step 1: Find the coordinates of point R
The given line L is in symmetric form. We can express the coordinates of any point on the line in terms of a parameter λ. Since R(5, p, q) lies on the line, we can substitute its x -coordinate to find λ and then determine p and q.
Step 2: Find the foot of the perpendicular from Q to L
Let M be the foot of the perpendicular from Q to the line L. Any point on L can be written as (2λ+1,3λ−1,4λ). The vector QM is perpendicular to the direction vector of the line L, which is b=(2,3,4). Their dot product must be zero.
Step 3: Determine the coordinates of M and P
Solving the dot product equation for λ, we find λ=1. Substituting this value back into the general point on the line gives the coordinates of M. Since P is the image of Q in the line L, M is the midpoint of PQ. We can use the midpoint formula to find the coordinates of P.
Step 4: Calculate the area of triangle PQR
Now that we have the coordinates of P, Q, and R, we can calculate the area of △PQR. The area of a triangle with vertices P, Q, R can be found using half the magnitude of the cross product of two vectors forming two sides of the triangle, for example, RP and RQ.
Step 5: Compute the cross product and its magnitude
We compute the cross product of RP and RQ. Then, we find the magnitude of this resultant vector. This magnitude is twice the area of the triangle.
Step 6: Calculate the square of the area
Finally, we calculate the area of the triangle and then square the result to get the required answer.