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JEE MainMathematicsThree Dimensional Geometry

A line L passing through the point P(4, 4, 7) intersects the lines L₁ : x-1 2 = y-2 1 = z-3 2 and L₂ : x-2 1 = y-3 2 = z-4 3 . If d is the perpendicular distance from the point C(1, -2, 3) to the line L , then the value of 6d^2 is equal to ________.

Correct answer

77

Step-by-step solution

Let the line L intersect L₁ at M(1+2 , 2+ , 3+2 ) and L₂ at N(2+ , 3+2 , 4+3 ) . Since P(4, 4, 7) , M , and N are collinear, the direction ratios of PM and PN must be proportional. Direction ratios of PM : 2 -3, -2, 2 -4 Direction ratios of PN : -2, 2 -1, 3 -3 Thus, 2 -3 -2 = -2 2 -1 = 2 -4 3 -3 . Taking the last two ratios: -2 2 -1 = 2( -2) 3( -1) ( -2)(3 -3) = 2( -2)(2 -1) ( -2)(3 - 3 - 4 + 2) = 0 ( -2)(- - 1) = 0 . So, either = 2 or = -1 . If = 2 , the direction ratios of PM become 1, 0, 0 . For PN to be proport

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