Question

If \(A(3,-2,2)\), \(B(2,\lambda+1,5)\) are the end points of the diameter of a circle and if the point \((5,6,-1)\) lies on the circle, then \(\lambda =\)

• A. 6
• B. 8
• C. 7
• D. 5

Hint:

For diameter endpoints, always apply the right-angle property using dot product.

Answer 1

The Correct Option is B

Step 1: Use the property of a circle.
If a point lies on a circle whose diameter endpoints are \(A\) and \(B\), then \[ \angle APB = 90^\circ \] which implies \[ \vec{PA} \cdot \vec{PB} = 0 \]
Step 2: Find vectors \(\vec{PA}\) and \(\vec{PB}\).
\[ \vec{PA} = (3-5,\,-2-6,\,2+1) = (-2,-8,3) \] \[ \vec{PB} = (2-5,\,\lambda+1-6,\,5+1) = (-3,\lambda-5,6) \]
Step 3: Take dot product.
\[ (-2)(-3) + (-8)(\lambda-5) + (3)(6) = 0 \] \[ 6 - 8\lambda + 40 + 18 = 0 \Rightarrow 64 - 8\lambda = 0 \]
Step 4: Solve for \(\lambda\).
\[ \lambda = 8 \]