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_________

Find the equation of the line passing through the points (1,2,3)a n d(-1,0,4)dot

In a triangle ABC, coordinates of A are (1,2) and the equations of the medians through B and C are x+y=5 and x=4 respectively. Find the coordinates of B and C.

The points A,B,C,D have the respective coordinates (-2,-3),(6,-5) , (18,9) and (0,12) . Find the area of the quadrilateral ABCD.

If C_1,C_2,a n dC_3 belong to a family of circles through the points (x_1,y_1)a n d(x_2, y_2) prove that the ratio of the length of the tangents from any point on C_1 to the circles C_2a n dC_3 is constant.

The line joining the points (-2,1,-8)a n d(a ,b ,c) is parallel to the line whose direction ratios are 6,2,a n d3. Find the values of a ,b and c

A straight line through the origin 'O' meets the parallel lines 4x +2y= 9 and 2x +y=-6 at points P and Q respectively. Then the point 'O' divides the segment PQ in the ratio : (A) 1:2 (B) 3:2 (C) 2:1 D) 4:3

A circle of radius 1 unit touches the positive x-axis and the positive y-axis at A and B , respectively. A variable line passing through the origin intersects the circle at two points D and E . If the area of triangle D E B is maximum when the slope of the line is m , then find the value of m^(-2)

The equations of the perpendicular bisectors of the sides A Ba n dA C of triangle A B C are x-y+5=0 and x+2y=0 , respectively. If the point A is (1,-2) , then find the equation of the line B Cdot

Let C be a curve defined by y=e^(a+b x^2)dot The curve C passes through the point P(1,1) and the slope of the tangent at P is (-2)dot Then the value of 2a-3b is_____.

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