If c=a+b, then show that the curves `x^((2)/(3))+y^((2)/(3))=c^((2)/(3))` and `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` touch each other.

If x : a = y : b = z : c , then prove that x^(3)/a^(2) + y^(3)/b^(2) + z^(3)/c^(2) = ((x+y+z)^(3))/((a+b+c)^(2))

If a : b = x : y , then show that (a^(2) + b^(2)) : a^(3)/(a + b) : : (x^(2) + y^(2)) : x^(3)/(x + y)

If a/b = x/y , then show that (a+b) (a^(2)+b^(2))x^(3) = (x+y)(x^(2)+y^(2))a^(3)

Show that the length of the common chord of the circles (x-a)^(2) + (y-b)^(2) = c^(2) and (x-b)^(2) + (y-a)^(2) = c^(2) is sqrt(4c^(2) -2(a-b)^(2)) unit.

Find the angle between the curves x^2-(y^2)/3=a^2a n dC_2: x y^3=c

If the chord of contact of tangents from a point on the circle x^(2) + y^(2) = a^(2) to the circle x^(2)+ y^(2)= b^(2) touches the circle x^(2) + y^(2) = c^(2) , then a, b, c are in-

Show that the curve xy=a^2 and x^2+y^2=2a^2 touch each other.

The straight lines (x)/(a_(1))=(y)/(b_(1))=(z)/(c_(1)) and (x-2)/(a_(2))=(y-3)/(b_(2))=(z)/(c_(2)) will be parallel if -

If the circles x^(2) + y^(2) + 2ax + c^(2) = 0 and x^(2) + y^(2) + 2by + c^(2) = 0 touch each other, prove that, (1)/(a^(2)) + (1)/(b^(2)) = (1)/(c^(2)) .

If (x-a)^(2)+(y-b)^(2)=r^(2) , show that, ((1+y_(1)^(2))^((3)/(2)))/(y_(2))=-r .