If the quadrilateral formed by the lines `a x+b y+c=0,a^(prime)x+b^(prime)y+c=0,a x+b y+c^(prime)=0,a^(prime)x+b^(prime)y+c^(prime)=0` has perpendicular diagonals, then (a) `b^2+c^2=b^('2)+c^('2)` (b)`c^2+a^2=c^('2)+a^('2)` (c)`a^2+b^2=a^('2)+b^('2)` (d) none of these

Fid the condition if lines x=a y+b ,z=c y+da n dx=a^(prime)y+b^(prime), z=c^(prime)y+d ' are perpendicular.

Find the condition if lines x=a y+b ,z=c y+da n dx=a^(prime)y+b^(prime), z=c^(prime)y+d ' are perpendicular.

The line a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)=0 are perpendicular if:

If the circles x^2+y^2+2a^(prime)x+2b^(prime)y+c^(prime)=0 and 2x^2+2y^2+2a x+2b y+c=0 intersect othrogonally, then prove that a a^(prime) + b b prime=c+c^(prime)/2dot

If the circle x^2+y^2+2gx+2fy+c=0 bisects the circumference of the circle x^2+y^2+2g^(prime)x+2f^(prime)y+c^(prime)=0 then prove that 2g^(prime)(g-g^(prime))+2f^(prime)(f-f^(prime))=c-c '

The area of a parallelogram formed by the lines a x+-b x+-c=0 is (a) (c^2)/((a b)) (b) (s c^2)/((a b)) (c) (c^2)/(2a b) (d) none of these

If (a x^2+c)y+(a^(prime)x^2+c^(prime) )=0 and x is a rational function of y and ac is negative, then a. a c^(prime)+c^(prime)c=0 b. a//a '=c//c ' c. a^2+c^2=a^('2)+c^('2) d. a a^(prime)+c c^(prime)=1

The condition that one of the straight lines given by the equation a x^2+2h x y+b y^2=0 may coincide with one of those given by the equation a^(prime)x^2+2h^(prime)x y+b^(prime)y^2=0 is (a b^(prime)-a^(prime)b)^2=4(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (a b^(prime)-a^(prime)b)^2=(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (h a^(prime)-h^(prime)a)^2=4(a b^(prime)-a^(prime)b)(b h^(prime)-b^(prime)h) (b h^(prime)-b^(prime)h)^2=4(a b^(prime)-a^(prime)b)(h a^(prime)-h^(prime)a)

If the differential equation representing the family of curves y=C_1 cos 2 x+C_2 cos ^2 x+C_3 sin ^2 x+C_4 is lambda y^(prime)=y^(prime prime) tan2 x , then lambda is

The line A x+B y+C=0 cuts the circle x^2+y^2+a x+b y+c=0 at Pa n dQ . The line A^(prime)x+B^(prime)x+C^(prime)=0 cuts the circle x^2+y^2+a^(prime)x+b^(prime)y+c^(prime)=0 at Ra n dSdot If P ,Q ,R , and S are concyclic, then show that |a-a ' b-b ' c-c ' A B C A ' B ' C '|=0