`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`

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 (c) 45^0 (d) 90^0

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

Two circles C_1a n dC_2 intersect at two distinct points Pa n dQ in a line passing through P meets circles C_1a n dC_2 at Aa n dB , respectively. Let Y be the midpoint of A B ,a n dQ Y meets circles C_1a n dC_2 at Xa n dZ , respectively. Then prove that Y is the midpoint of X Zdot

If a ,b , a n dc are in G.P. and x ,y , respectively, are the arithmetic means between a ,b ,a n db ,c , then the value of a/x+c/y is 1 b. 2 c. 1//2 d. none of these

If three points are A(-2,1)B(2,3),a n dC(-2,-4) , then find the angle between A Ba n dB Cdot

The exradii r_1,r_2 and r_3 of /_\A B C are in H.P. Show that its sides a , ba n dc are in AdotPdot

If a plane meets the equations axes at A ,Ba n dC such that the centroid of the triangle is (1,2,4), then find the equation of the plane.

If a plane meets the equations axes at A ,Ba n dC such that the centroid of the triangle is (1,2,4), then find the equation of the plane.

Two circles C_1a n dC_2 both pass through the points A(1,2)a n dE(2,1) and touch the line 4x-2y=9 at Ba n dD , respectively. The possible coordinates of a point C , such that the quadrilateral A B C D is a parallelogram, are (a ,b)dot Then the value of |a b| is_________

Let a given line L_1 intersect the X and Y axes at P and Q respectively. Let another line L_2 perpendicular to L_1 cut the X and Y-axes at Rand S, respectively. Show that the locus of the point of intersection of the line PS and QR is a circle passing through the origin