If the lines \(2x - 3y = 5\) and \(3x - 4y = 7\) are the diameters of a circle of area 154 square units, then obtain the equation of the circle.
\(x^2 + y^2 - 2x + 2y = 47\)
Find the equation of the circle which passes through the points \((2,3)\) and \((4,5)\) and the centre lies on the straight line \(y - 4x + 3 = 0\).
\(x^2 + y^2 - 4x - 10y + 25 = 0\)
Find the equation of a circle whose centre is \((3,-1)\) and which cuts off a chord of length 6 units on the line \(2x - 5y + 18 = 0\).
\((x - 3)^2 + (y + 1)^2 = 38\)
Find the equation of a circle of radius 5 which is touching another circle \(x^2 + y^2 - 2x - 4y - 20 = 0\) at \((5,5)\).
\(x^2 + y^2 - 18x - 16y + 120 = 0\)
Find the equation of a circle passing through the point \((7,3)\) having radius 3 units and whose centre lies on the line \(y = x - 1\).
\(x^2 + y^2 - 8x - 6y + 16 = 0\)
Find the equation of each of the following parabolas:
(a) Directrix \(x = 0\), focus at \((6,0)\)
(b) Vertex at \((0,4)\), focus at \((0,2)\)
(c) Focus at \((-1,-2)\), directrix \(x - 2y + 3 = 0\)
(a) \(y^2 = 12x - 36\)
(b) \(x^2 = 32 - 8y\)
(c) \(4x^2 + 4xy + y^2 + 4x + 32y + 16 = 0\)
Find the equation of the set of all points the sum of whose distances from the points \((3,0)\) and \((9,0)\) is 12.
\(3x^2 + 4y^2 - 36x = 0\)
Find the equation of the set of all points whose distance from \((0,4)\) are \(\tfrac{2}{3}\) of their distance from the line \(y = 9\).
\(9x^2 + 5y^2 = 180\)
Show that the set of all points such that the difference of their distances from \((4,0)\) and \((-4,0)\) is always equal to 2 represent a hyperbola.
\(15x^2 - y^2 = 15\)
Find the equation of the hyperbola with
(a) Vertices \((\pm5,0)\), foci \((\pm7,0)\)
(b) Vertices \((0,\pm7)\), eccentricity \(\dfrac{4}{3}\)
(c) Foci \((0,\pm\sqrt{10})\), passing through \((2,3)\)
(a) \(15x^2 - y^2 = 15\)
(b) \(9x^2 - 7y^2 + 343 = 0\)
(c) \(y^2 - x^2 = 5\)