If A $(3,-2,2), \mathrm{B}(2, \lambda+1,5)$ are the end points of the diameter of the circle and if the…
If A $(3,-2,2), \mathrm{B}(2, \lambda+1,5)$ are the end points of the diameter of the circle and if
the point $(5,6,-1)$ lies on the circle, then $\lambda=$
6
8
7
5
Solution
The angle subtended at the point $\mathrm{P}$ in the semicircle APB is a right angle. AP $\perp P B$
Now direction ratios of AP are $(5-3,6+2,-1-2)$ i.e. $(2,8,-3)$
Now direction ratios of PB are $(2-5, \lambda+1-6,5+1)$ i.e. $(-3, \lambda-5,6)$
Since AP $\perp P B$, we write
(2) $(-3)+8(\lambda-5)-3(6)=0 \Rightarrow-6+8 \lambda-40-18=0 \Rightarrow \lambda=8$
