Let $${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}},\,a,b,c \in R$$, represents a circle with center ($$\alpha$$, $$\beta$$). Then, $$\alpha$$ + 2$$\beta$$ is equal to :

Let $$\alpha$$₁, $$\alpha$$₂ ($$\alpha$$₁ < $$\alpha$$₂) be the values of $$\alpha$$ fo the points ($$\alpha$$, $$-$$3), (2, 0) and (1, $$\alpha$$) to be collinear. Then the equation of the line, passing through ($$\alpha$$₁, $$\alpha$$₂) and making an angle of $${\pi \over 3}$$ with the positive direction of the x-axis, is :

Consider three circles:

$${C_1}:{x^2} + {y^2} = {r^2}$$

$${C_2}:{(x - 1)^2} + {(y - 1)^2} = {r^2}$$

$${C_3}:{(x - 2)^2} + {(y - 1)^2} = {r^2}$$

If a line L : y = mx + c be a common tangent to C₁, C₂ and C₃ such that C₁ and C₃ lie on one side of line L while C₂ lies on other side, then the value of $$20({r^2} + c)$$ is equal to :

Let the eccentricity of the ellipse $${x^2} + {a^2}{y^2} = 25{a^2}$$ be b times the eccentricity of the hyperbola $${x^2} - {a^2}{y^2} = 5$$, where a is the minimum distance between the curves y = e^x and y = log_ex. Then $${a^2} + {1 \over {{b^2}}}$$ is equal to :