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 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 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 '}

The determinant |y^2-x y x^2a b c a ' b ' c '| is equal to a. |b x+a y c x+b y b^(prime)x+a ' y c^(prime)x+b ' y| b. |a x+b y b x+c y a^(prime)x+b ' y b ' x+c ' y| c. |b x+c y a x+b y b^(prime)x+c ' y a^(prime)x+b ' y| d. |a x+b y b x+c y a^(prime)x+b ' y b^(prime)x+c ' y|