Prove that the system of equations in `x\ a n d\ y ; a x_(h y)+g=0,\ h x+b y+f=0,\ a x^2+2h x y+b y^2+2gx+2fy+c=t` is consistent if `t=|a hgh bfgfc|-:|a h h b|`

If c lt 1 and the system of equations x+y-1=0 2x-y-c=0 and -bx+3by-c =0 is consistent then the possible real values of b are

Obtain equation of circle in x^(2) + y^(2) - x + y = 0

If the system of linear equations x+a y+a z=0,\ x+b y+b z=0,\ x+c y+c z=0 has a non zero solution then System is always non trivial solutions System is consistent only when a=b=c If a!=b!=c then x=0,\ y=t ,\ z=-t\ AAt in R If a=b=c\ t h e n\ y=t_1, z=t_2,\ x=-a(t_1+t_2)AAt_1, t_2 in R

If the system of equations x+y-3=0,\ (1+K)x+(2+K)y-8=0\ &\ x-(1+K)y+(2+K)=0 is consistent then the value of K may be - a. 1 b. 3/5 c. -5/3 d. 2

Prove that the centres of the circle x^(2) + y^(2) - 4x - 2y + 4 = 0, x^(2) + y^(2) - 2x - 4y + 1 = 0 and x^(2) + y^(2) + 2x - 8y + 1 = 0 are collinear. More over prove that their radii are in geometric pregression.

A point moves so that the distance between the foot of perpendiculars from it on the lines a x^2+2hx y+b y^2=0 is a constant 2d . Show that the equation to its locus is (x^2+y^2)(h^2-a b)=d^2{(a-b)^2+4h^2}dot

Obtain the area of equilateral triangle inscribed in circle x^(2) + y^(2) + 2gx + 2fy + c = 0 .

If the circle x^(2) + y^(2) + 2gx + 2fy + c = 0 does not intersects the X - axis then ……..