Prove that quadrilateral `A B C D` , where `A B=x+y-10 ,B C=x-7y+50=0,C D=22 x-4y+125=0, and
DA =2x-4y-5=0,` is concyclic. Also find the equation of the circumcircle of `A B C Ddot`

Show that a cyclic quadrilateral is formed by the lines 5x+3y=9,x=3y ,2x=y and x+4y+2=0 taken in order. Find the equation of the circumcircle.

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

In triangle A B C , the equation of the right bisectors of the sides A B and A C are x+y=0 and y-x=0 , respectively. If A-=(5,7) , then find the equation of side B Cdot

In triangle A B C , the equation of side B C is x-y=0. The circumcenter and orthocentre of triangle are (2, 3) and (5, 8), respectively. The equation of the circumcirle of the triangle is a) x^2+y^2-4x+6y-27=0 b) x^2+y^2-4x-6y-27=0 c) x^2+y^2+4x-6y-27=0 d) x^2+y^2+4x+6y-27=0

Prove that (a b+x y)(a x+b y)>4a b x y(a , b ,x ,y >0)dot

If the quadrilateral formed by the lines ax + by + c = 0, a'x + b'y + c' = 0, ax + by + c' = 0,a'x + b'y + c = 0 have perpendicular diagonals, then :

In a triangle A B C , if A is (2,-1),a n d7x-10 y+1=0 and 3x-2y+5=0 are the equations of an altitude and an angle bisector, respectively, drawn from B , then the equation of B C is (a) a+y+1=0 (b) 5x+y+17=0 (c) 4x+9y+30=0 (d) x-5y-7=0

A line through A(-5,-4) meets the lines x+3y+2=0, 2x+y+4=0 and x-y-5=0 at the points B, C and D respectively. If (15/(AB))^2+(10/(AC))^2=(6/(AD))^2 find the equation of the line.

The lines parallel to the normal to the curve x y=1 is/are (a) 3x+4y+5=0 (b) 3x-4y+5=0 (c) 4x+3y+5=0 (d) 3y-4x+5=0

Let A B C be a triangle. Let A be the point (1,2),y=x be the perpendicular bisector of A B , and x-2y+1=0 be the angle bisector of /_C . If the equation of B C is given by a x+b y-5=0 , then the value of a+b is (a) 1 (b) 2 (c) 3 (d) 4