Question
If x/1 = y /1 = z/-1 represents one of the three mutually perpendicular generators of the cone 3xy + 8xz - 5yz = 0 , find the equations of the other two.
Answer :
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We are given that the equation of the cone is: \[ 3xy + 8xz - 5yz = 0 \] We are also given that one of the mutually perpendicular generators has the direction ratios given by: \[ \frac{x}{1} = \frac{y}{1} = \frac{z}{-1} \] From this, we can extract the direction ratios of the given generator as: \[ (1, 1, -1) \] ### Step 1: Finding the Other Two Mutually Perpendicular Generators Since the generators are mutually perpendicular, the direction ratios of the other two generators must satisfy: 1. They must lie on the given cone equation. 2. They must be perpendicular to each other as well as to \((1,1,-1)\). Let the direction ratios of the second generator be \((a, b, c)\) and of the third generator be \((p, q, r)\). #### Condition 1: Satisfying the Cone Equation For a cone equation of the form:
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