If two circles `x^(2)+y^(2)+2a_(1)x+2b_(1)y=0` and `x^(2)+y^(2)+2a_(2)x+2b_(2)y=0` touches then show that `a_(1)b_(2)=a_(2)b_(1)`

The circles x^(2)+y^(2)+2x-2y+1=0 and x^(2)+y^(2)-2x-2y+1=0 touch each other

If two equation a_(1) x^(2) + b_(1) x + c_(1) = 0 and, a_(2) x^(2) + b_(2) x + c_(2) = 0 have a common root, then the value of (a_(1) b_(2) - a_(2) b_(1)) (b_(1) c_(2) - b_(2) c_(1)) , is

Prove that the circle x^(2)+y^(2)+2x+2y+1=0 and circle x^(2)+y^(2)-4x-6y-3=0 touch each other.

Show that the circle passing through the origin and cutting the circles x^2+y^2-2a_1x-2b_1y+c_1=0 and x^2+y^2-2a_2x -2b_2y+c_2=0 orthogonally is |(x^2+y^2,x,y),(c_1,a_1,b_1),(c_2,a_2,b_2)|=0

Show that the circles x^(2) + y^(2) + 2x = 0 and x^(2)+ y^(2) - 6 x -6 y + 2 = 0 touch externally at the point (1/5,3/5)

Show that the lines a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)=0,"where" b_(1),b_(2) ne 0 "are (i) parallel, if"(a_(1))/(b_(1))=(a_(2))/(b_(2))" (ii) perpendicular, if "a_(1)a_(2)+b_(1)b_(2)=0

Statement 1 : If two conics a_(1)x^(2)+ 2h_(1)xy+b_(1)^(2)=c_(1) , a_(2)x^(2) +2h_(2) xy +b_(2)y^(2) =c_(2) intersect in 4 concyclic points, then ( a_(1) -b_(1)) h_(2)=(a_(2)-b_(2))h_(1) . Statement 2 : For a conic to be a circle, coefficient of x^(2) = coefficient of y^(2) and coefficient of xy =0.

If the lines x=a_(1)y + b_(1), z=c_(1)y +d_(1) and x=a_(2)y +b_(2), z=c_(2)y + d_(2) are perpendicular, then

If the tangent at point P(h, k) on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 cuts the circle x^(2)+y^(2)=a^(2) at points Q(x_(1),y_(1)) and R(x_(2),y_(2)) , then the value of (1)/(y_(1))+(1)/(y_(2)) is

STATEMENT-1: If three points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) are collinear, then |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 STATEMENT-2: If |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 then the points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) will be collinear. STATEMENT-3: If lines a_(1)x+b_(1)y+c_(1)=0,a_(2)=0and a_(3)x+b_(3)y+c_(3)=0 are concurrent then |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0