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 '`

If f(x)=x|x|, then prove that f^(prime)(x)=2|x|

If f(x)=x|x|, then prove that f^(prime)(x)=2|x|

If f(x)=x|x|, then prove that f^(prime)(x)=2|x|

If the circle x^(2) +y^(2) + 2gx + 2fy + c = 0 bisects the circumference of the circle x^(2) +y^(2) + 2g'x + 2f'y + c' = 0 , them the length of the common chord of these two circles is

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)

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 f(x)=(x+3)/(5x^2+x-1) and g(x)=(2x+3x^2)/(20+2x-x^2) such that f(x) and g(x) are differentiable functions in their domains, then which of the following is/are true (a) 2f^(prime)(2)+g^(prime)(1)=0 (b) 2f^(prime)(2)-g^(prime)(1)=0 (c) f^(prime)(1)+2g^(prime)(2)=0 (d) f^(prime)(1)-2g^(prime)(2)=0

Show that the condition that the circle x^2+y^2+2g_1x+2f_1y+c_1=0 bisects the circumference of the circle x^2+y^2+2g_2x+2f_2y+c_2=0 is 2(g_1-g_2)g_2+2(f_1-f_2)f_2=c_1-c_2

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

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