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`

Two circles C_1 and C_2 intersect at two distinct points P and Q in a line passing through P meets circles C_1 and C_2 at A and B , respectively. Let Y be the midpoint of A B and Q Y meets circles C_1 and C_2 at X a n d Z respectively. Then prove that Y is the midpoint of X Z

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

In an isosceles triangle A B C with A B=A C , a circle passing through B and C intersects the sides A Ba n dA C at Da n dE respectively. Prove that DE∣∣BC Bdot

A line passing through the origin O(0,0) intersects two concentric circles of radii aa n db at Pa n d Q , If the lines parallel to the X-and Y-axes through Qa n dP , respectively, meet at point R , then find the locus of Rdot

If a ,b ,c are in G.P. and x ,y are the arithmetic means of a ,ba n db ,c respectively, then prove that a/x+c/y=2a n d1/x+1/y=2/b

Two circles with centres Aa n dB of radii 3c ma n d4c m respectively intersect at two points Ca n dD such that A Ca n dB C are tangents to the two circles. Find the length of the common chord C Ddot

A point P moves on a plane x/a+y/b+z/c=1. A plane through P and perpendicular to O P meets the coordinate axes at A , Ba n d Cdot If the planes through A ,Ba n dC parallel to the planes x=0,y=0a n dz=0, respectively, intersect at Q , find the locus of Qdot

A point P moves on a plane x/a+y/b+z/c=1. A plane through P and perpendicular to O P meets the coordinate axes at A , Ba n d Cdot If the planes through A ,Ba n dC parallel to the planes x=0,y=0a n dz=0, respectively, intersect at Q , find the locus of Qdot

In Figure, Pa n dQ are the midpoints of the sides C Aa n dC B respectively of A B C right angled at C. Prove that 4(A Q^2+B P^2)=5A B^2 .

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 B and D , 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_________