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)+bb^(prime)=c+c^(prime/)2dot`

If the circles x^(2)+y^(2)+2a'x+2b'y+c'=0 and 2x^(2)+2y^(2)+2ax+2by+c=0 intersect othrogonally,then prove that aa'+bbquad =c+(c')/(2)

If curves (x^2)/(a^2)-(y^2)/(b^2)=1 and xy= c^2 intersect othrogonally , then

Show the condition that the curves a x^2+b y^2=1 and a^(prime)\ x^2+b^(prime)\ y^2=1 Should intersect orthogonally

If the abscissae and the ordinates of two point Aa n dB be the roots of a x^2+b x+c=0a n da^(prime)y^2+b^(prime)y=c^(prime)=0 respectively, show that the equation of the circle described on A B as diameter is a a^(prime)(x^2+y^2)+a^(prime)b x+a b^(prime)y+(c a^(prime)+ca)=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 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 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

If f(x)=|x+a^2a b a c a b x+b^2b c a c b c x+c^2|,t h e n prove that f^(prime)(x)=3x^2+2x(a^2+b^2+c^2)dot

If A=[[3,4],[ -1,2],[0, 1]] and B=[[-1, 2, 1],[1,2, 3]] , then verify that (A+B)^(prime)=A^(prime)+B^(prime)