In a triangle $A B C$ , right angled at the vertex A, if the position vectors of $A, B$ and $C$ are respectively $3^i +^j -^k, -^i + 3^j + p^k$ and $5^i + q^j - 4^k$ , then the point $(p, q)$ lies on a line:

Making an obtuse angle with the positive direction of x-axis

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Making an acute angle with the positive direction of x-axis

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Parallel to x-axis

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Parallel to y-axis

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Solution

The correct option is B Making an acute angle with the positive direction of x-axis
Triangle ABC is right angled at A. So Line BA and CA are perpendicular to each other.
Now, Vector Equation of line
BA $= (- i + 3 j + p k) - (3 i + j - k) = [- 4 i + 2 j + (p + 1) k]$
CA $= (5 i + q j - 4 k) - (3 i + j - k) = [2 i + (q - 1) j - 3 k]$
Since Line BA and CA are perpendicular to each other so BA $.$ CA $= 0$
So, $- 8 + 2 q - 2 - 3 p - 3 = 0$
$∴ 2 q = 3 p + 13$
So we can say $(p, q)$ lies on the line $2 y = 3 x + 13$
And slope of this line is positive so it makes an acute angle with the positive direction of x-axis

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In a triangle $A B C,$ right angled at the vertex $A,$ if the position vectors of $A, B$ and $C$ are respectively $3^i +^j -^k, -^i + 3^j + p^k$ and $5^i + q^j - 4^k$ then the point $(p, q)$ lies on a line

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In a triangle ABC, right angled at the vertex A, if the position vectors of A,B and C are respectively 3^i+^j−^k,−^i+3^j+p^k and 5^i+q^j−4^k, then the point (p,q) lies on a line:

In a triangle $A B C$ , right angled at the vertex A, if the position vectors of $A, B$ and $C$ are respectively $3^i +^j -^k, -^i + 3^j + p^k$ and $5^i + q^j - 4^k$ , then the point $(p, q)$ lies on a line: