Let the point A divide the line segment joining the points P(-1,-1,2) and Q(5,5,10) internally in the ratio r: 1(r>0). If O is the origin and (OQ⃗ · OA⃗)-1/5|OP⃗ × OA⃗|²=10, then the value of r is :

Solution

Maths sir · 5 steps

Write OA in terms of OP and OQ, and then OP × OA is just a multiple of OP × OQ.

1. Section formula. A divides PQ in the ratio r:1, so OA⃗ = (OP⃗ + r OQ⃗)/(r+1) = 1/(r+1)(5r-1, 5r-1, 10r+2).
2. Dot product. OQ⃗ · OA⃗ = (5(5r-1) + 5(5r-1) + 10(10r+2))/(r+1) = (150r + 10)/(r+1).
3. Cross product. Since OP⃗ × OP⃗ = 0⃗, you get OP⃗ × OA⃗ = r/(r+1)(OP⃗ × OQ⃗). Here OP⃗ × OQ⃗ = (-20, 20, 0), so its square magnitude is 800, and |OP⃗ × OA⃗|² = 800r²/(r+1)².

   (150r+10)/(r+1) - 160r²/(r+1)² = 10

4. Clear denominators. Multiply by (r+1)². This gives -10r² + 160r + 10 = 10r² + 20r + 10, so 20r² - 140r = 0, which means 20r(r - 7) = 0.
5. So. Since r > 0, the root r = 0 is ruled out, and r = 7, which is option (c).

The idea to keep. When a point lies on a segment, write its position vector as a mix of the end points. A cross product with one end point then wipes that part out, and the rest is a simple scalar multiple.

Answer from the NTA final answer key. Solution by PrepWiser.