`P_1n dP_2` are planes passing through origin `L_1a n dL_2` are two lines on`P_1a n dP_2,` respectively, such that their intersection is the origin. Show that there exist points `A ,Ba n dC ,` whose permutation `A^(prime),B^(prime)a n dC^(prime),` respectively, can be chosen such that `A` is on `L_1,BonP_1` but not on `L_1a n dC` not on `P_1;` `A '` is on `L_2,B 'onP_2` but not on `L_2a n dC '` not on `P_2dot`

A plane meets the coordinate axes at A ,Ba n dC respectively such that the centroid of triangle A B C is (1,-2,3)dot Find the equation of the plane.

Two planes P_1 and P_2 pass through origin. Two lines L_1 and L_2 also passingthrough origin are such that L_1 lies on P_1 but not on P_2, L_2 lies on P_2 but not on P_1 A,B, C are there points other than origin, then prove that the permutation [A', B', C'] of [A, B, C] exists. Such that: (a) A lies on L1, B lies on P1 not on L1, C does not lie on P1 . (b) A lies on L2, B lies on P2 not on L2, C' does not lies on P2.

L_1a n dL_2 are two lines. If the reflection of L_1onL_2 and the reflection of L_2 on L_1 coincide, then the angle between the lines is (a) 30^0 (b) 60^0 45^0 (d) 90^0

Two systems of rectangular axes have the same origin. If a plane cuts them at distances a ,b ,ca n da^(prime),b^(prime),c^(prime) respectively, prove that 1/(a^2)+1/(b^2)+1/(c^2)=1/(a^('2))+1/(b^('2))+1/(c^('2))

Two systems of rectangular axes have the same origin. If a plane cuts them at distances a ,b ,ca n da^(prime),b^(prime),c^(prime) respectively, prove that 1/(a^2)+1/(b^2)+1/(c^2)=1/(a^('2))+1/(b^('2))+1/(c^('2))

Determine the point in X Y-p l a n e whcih is equidistant from three points A(2,0,3),B(0,3,2)a n dC(0,0,1)dot

Find the equation of the altitudes of the triangle whose vertices are A(7,-1),B(-2,8)a n dC(1,2)dot

Find the equations to the altitudes of the triangle whose angular points are A(2,-2),B(1,1)a n dC(-1,0)dot

The chords of contact of tangents from three points A ,Ba n dC to the circle x^2+y^2=a^2 are concurrent. Then A ,B a n dC will be

Consider two lines L_1a n dL_2 given by a_1x+b_1y+c_1=0a n da_2x+b_2y+c_2=0 respectively where c1 and c2 !=0, intersecting at point PdotA line L_3 is drawn through the origin meeting the lines L_1a n dL_2 at Aa n dB , respectively, such that PA=P B . Similarly, one more line L_4 is drawn through the origin meeting the lines L_1a n dL_2 at A_1a n dB_2, respectively, such that P A_1=P B_1dot Obtain the combined equation of lines L_3a n dL_4dot