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 the circle x^(2)+y^(2)+2gx+2fy+c=0 bisects the circumference of the circles x^(2)+y^(2)=6,x^(2)+y^(2)-6x-4y+10=0 and x^(2)+y^(2)+2x+2y-2=0

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 a function is represented parametrically be the equations x=(1+(log)_e t)/(t^2); y=(3+2(log)_e t)/t , then which of the following statements are true? (a) y^('')(x-2x y^(prime))=y (b) y y^(prime)=2x(y^(prime))^2+1 (c) x y^(prime)=2y(y^(prime))^2+2 (d) y^('')(y-4x y^(prime))=(y^(prime))^2

The differential equation of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=C is a. y^(primeprime)/y^(prime)+y^(prime)/y-1/x=0 b. y^(primeprime)/y^(prime)+y^(prime)/y+1/x=0 c. y^(primeprime)/y^(prime)-y^(prime)/y-1/x=0 d. none of these

Consider the family of all circles whose centers lie on the straight line y=x . If this family of circles is represented by the differential equation P y^(primeprime)+Q y^(prime)+1=0, where P ,Q are functions of x , y and y^(prime)(h e r ey^(prime)=(dy)/(dx),y^=(d^2y)/(dx^2)), then which of the following statements is (are) true? (a)P=y+x (b)P=y-x (c)P+Q=1-x+y+y^(prime)+(y^(prime))^2 (d)P-Q=x+y-y^(prime)-(y^(prime))^2

If the equations x^2+p x+q=0a n dx^2+p^(prime)x+q^(prime)=0 have a common root, then it must be equal to a. (p^(prime)-p ^(prime) q)/(q-q^(prime)) b. (q-q ')/(p^(prime)-p) c. (p^(prime)-p)/(q-q^(prime)) d. (p q^(prime)-p^(prime) q)/(p-p^(prime))

The circle x^(2)+y^(2)+2gx+2fy+c=0 does not intersect x-axis,if (a) g^(2) c(c)g^(2)<2c(d) none of these

If alpha,beta are the roots of x^2-p x+q=0a n dalpha^(prime),beta' are the roots of x^2-p^(prime)x+q^(prime)=0, then the value of (alpha-alpha^(prime))^2+(beta-alpha^(prime))^2+(alpha-beta^(prime))^2+(beta-beta^(prime))^2 is 2{p^2-2q+p^('2)-2q^(prime)-p p '} 2{p^2-2q+p^('2)-2q^(prime)-q q '} 2{p^2-2q-p^('2)-2q^(prime)-p p '} 2{p^2-2q-p^('2)-2q^(prime)-q q '}