Question

A point \(P\) lies on a line through \(Q(1,-2,3)\) and is parallel to the line \(\frac{x}{1}=\frac{y}{4}=\frac{z}{5}\). If \(P\) lies on the plane \(2 x+3 y-4 z+22=0\), then segment \(P Q\) equals to
(A) \(\sqrt{42}\) units
(B) \(\sqrt{32}\) units
(C) 4 units
(D) 5 units

Answer 1

Ans: (A)
Hint: Let \(P(\lambda+1,4 \lambda-2,5 \lambda+3)\)
It lies on \(2 x+3 y-4 z+22=0\)
\(\therefore 2(\lambda+1)+3(4 \lambda-2)-4(5 \lambda+3)+22=0\)
\(\therefore 6 \lambda=6 \quad \Rightarrow \lambda=1\)
\(\therefore P=(2,2,8) \quad \therefore P Q=\sqrt{1^{2}+4^{2}+5^{2}}=\sqrt{42}\)

Answer 2

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