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Most Important Selected Qs for JEE AdvancedMathematicsThree Dimensional Geometry

P is a point on the plane a x+b y+c z=d . A point Q is taken on the line O P such O P . O Q=d^2 , then the locus of Q satisfies d(a x+b y+c z) x^2+y^2+z^2 is equal to

Correct answer

1

Step-by-step solution

Let P (x₁, y₁, z₁ ) be a point on a x+b y+c z=d ...(1) Then ax ₁+ by ₁+ cz ₁= d Let O P=r . Then direction cosines of O P are x₁ r y₁ r , z₁ r Equation of lien OP is x-0 x₁ r = y-0 y₁ r = z-0 z₁ Let Q ( , , r) be a point on OP such that OQ = . Then coordinates of Q ( x₁ r , y₁ r , z₁ r ) . But coordinate of Q be ( , , r ) aligned & x₁= r , y₁= r , z₁= r^2 & a x₁+b y₁+c z₁=d & r (a +b +c r)=d...(ii) & O P O Q=d^2....(iii) & (a +b +c )=r=d & (a +b +c )=r =d ^2 & (a +b +c )=d^2=d ^2 & (a +b +c )=d= ^2= ^2+ ^2+ ^2 & l

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