Find the value of p such that the lines

\(\frac{x-1}{3}=\frac{y+4}{p}=\frac{z-9}{1}\)

\(\frac{x+2}{1}=\frac{y-3}{1}=\frac{z-7}{-2}\)

are at right angles to each other.

(a) p=2

(b) p=1

(c) p=-1

(d) p=-2

I got this question during an online exam.

This intriguing question originated from Three Dimensional Geometry in portion Three Dimensional Geometry of Mathematics – Class 12

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Right answer is (c) p=-1

Explanation: The angle between two lines is given by the equation

\(cosθ=\left |\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)

cos90°=\(\left |\frac{3(1)+p(1)+1(-2)}{\sqrt{3^2+p^2+1^2}.\sqrt{1^2+1^2+(-2)^2}}\right |\)

0=\(|\frac{p+1}{\sqrt{10+p^2}.√6}|\)

0=p+1

p=-1