Find the equation of the circle with centre \((0,2)\) and radius 2.
x^2 + y^2 - 4y = 0
Find the equation of the circle with centre \((-2,3)\) and radius 4.
x^2 + y^2 + 4x - 6y - 3 = 0
Find the equation of the circle with centre \(\left(\dfrac{1}{2}, \dfrac{1}{4}\right)\) and radius \(\dfrac{1}{12}\).
36x^2 + 36y^2 - 36x - 18y + 11 = 0
Find the equation of the circle with centre \((1,1)\) and radius \(\sqrt{2}\).
x^2 + y^2 - 2x - 2y = 0
Find the equation of the circle with centre \((-a,-b)\) and radius \(\sqrt{a^2 - b^2}\).
x^2 + y^2 + 2ax + 2by + 2b^2 = 0
Find the centre and radius of the circle \((x+5)^2 + (y-3)^2 = 36\).
c(-5, 3), r = 6
Find the centre and radius of the circle \(x^2 + y^2 - 4x - 8y - 45 = 0\).
c(2, 4), r = \(\sqrt{65}\)
Find the centre and radius of the circle \(x^2 + y^2 - 8x + 10y - 12 = 0\).
c(4, -5), r = \(\sqrt{53}\)
Find the centre and radius of the circle \(2x^2 + 2y^2 - x = 0\).
c\(\left(\dfrac{1}{4}, 0\right)\), r = \(\dfrac{1}{4}\)
Find the equation of the circle passing through the points (4,1) and (6,5) and whose centre lies on the line \(4x + y = 16\).
x^2 + y^2 - 6x - 8y + 15 = 0
Find the equation of the circle passing through the points (2,3) and (−1,1) and whose centre is on the line \(x - 3y - 11 = 0\).
x^2 + y^2 - 7x + 5y - 14 = 0
Find the equation of the circle with radius 5 whose centre lies on the x-axis and passes through the point (2,3).
x^2 + y^2 + 4x - 21 = 0 and x^2 + y^2 - 12x + 11 = 0
Find the equation of the circle passing through (0,0) and making intercepts \(a\) and \(b\) on the coordinate axes.
x^2 + y^2 - ax - by = 0
Find the equation of the circle with centre (2,2) and passing through the point (4,5).
x^2 + y^2 - 4x - 4y = 5
Does the point \((-2.5, 3.5)\) lie inside, outside or on the circle \(x^2 + y^2 = 25\)?
Inside the circle; since the distance of the point to the centre of the circle is less than the radius of the circle.
For the parabola \(y^2 = 12x\), find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.
F (3, 0), axis – x-axis, directrix \(x = -3\), length of latus rectum = 12
For the parabola \(x^2 = 6y\), find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.
F \((0, \tfrac{3}{2})\), axis – y-axis, directrix \(y = -\tfrac{3}{2}\), length of latus rectum = 6
For the parabola \(y^2 = -8x\), find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.
F (−2, 0), axis – x-axis, directrix \(x = 2\), length of latus rectum = 8
For the parabola \(x^2 = -16y\), find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.
F (0, −4), axis – y-axis, directrix \(y = 4\), length of latus rectum = 16
For the parabola \(y^2 = 10x\), find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.
F \((\tfrac{5}{2}, 0)\), axis – x-axis, directrix \(x = -\tfrac{5}{2}\), length of latus rectum = 10
For the parabola \(x^2 = -9y\), find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.
F \((0, -\tfrac{9}{4})\), axis – y-axis, directrix \(y = \tfrac{9}{4}\), length of latus rectum = 9
Find the equation of the parabola whose focus is (6, 0) and directrix is \(x = -6\).
\(y^2 = 24x\)
Find the equation of the parabola whose focus is (0, −3) and directrix is \(y = 3\).
\(x^2 = -12y\)
Find the equation of the parabola with vertex (0,0) and focus (3,0).
\(y^2 = 12x\)
Find the equation of the parabola with vertex (0,0) and focus (−2,0).
\(y^2 = -8x\)
Find the equation of the parabola with vertex (0,0) passing through (2,3) and whose axis is along the x-axis.
\(2y^2 = 9x\)
Find the equation of the parabola with vertex (0,0), passing through (5,2) and symmetric with respect to the y-axis.
\(2x^2 = 25y\)
In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
For the ellipse \(\dfrac{x^2}{36} + \dfrac{y^2}{16} = 1\).
Focus: \(F(\pm \sqrt{20}, 0)\)
Vertices: \(V(\pm 6, 0)\)
Major axis = \(12\)
Minor axis = \(8\)
Eccentricity: \(e = \dfrac{\sqrt{20}}{6}\)
Latus rectum = \(\dfrac{16}{3}\)
In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
For the ellipse \(\dfrac{x^2}{4} + \dfrac{y^2}{25} = 1\).
Focus: \(F(0, \pm \sqrt{21})\)
Vertices: \(V(0, \pm 5)\)
Major axis = \(10\)
Minor axis = \(4\)
Eccentricity: \(e = \dfrac{\sqrt{21}}{5}\)
Latus rectum = \(\dfrac{8}{5}\)
In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
For the ellipse \(\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1\).
Focus: \(F(\pm \sqrt{7}, 0)\)
Vertices: \(V(\pm 4, 0)\)
Major axis = \(8\)
Minor axis = \(6\)
Eccentricity: \(e = \dfrac{\sqrt{7}}{4}\)
Latus rectum = \(\dfrac{9}{2}\)
In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
For the ellipse \(\dfrac{x^2}{25} + \dfrac{y^2}{100} = 1\).
Focus: \(F(0, \pm \sqrt{75})\)
Vertices: \(V(0, \pm 10)\)
Major axis = \(20\)
Minor axis = \(10\)
Eccentricity: \(e = \dfrac{\sqrt{3}}{2}\)
Latus rectum = \(5\)
In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
For the ellipse \(\dfrac{x^2}{49} + \dfrac{y^2}{36} = 1\).
Focus: \(F(\pm \sqrt{13}, 0)\)
Vertices: \(V(\pm 7, 0)\)
Major axis = \(14\)
Minor axis = \(12\)
Eccentricity: \(e = \dfrac{\sqrt{13}}{7}\)
Latus rectum = \(\dfrac{72}{7}\)
In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
For the ellipse \(\dfrac{x^2}{100} + \dfrac{y^2}{400} = 1\).
Focus: \(F(0, \pm 10\sqrt{3})\)
Vertices: \(V(0, \pm 20)\)
Major axis = \(40\)
Minor axis = \(20\)
Eccentricity: \(e = \dfrac{\sqrt{3}}{2}\)
Latus rectum = \(10\)
In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
For the ellipse \(36x^2 + 4y^2 = 144\).
Focus: \(F(0, \pm 4\sqrt{2})\)
Vertices: \(V(0, \pm 6)\)
Major axis = \(12\)
Minor axis = \(4\)
Eccentricity: \(e = \dfrac{2\sqrt{2}}{3}\)
Latus rectum = \(\dfrac{4}{3}\)
In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
For the ellipse \(16x^2 + y^2 = 16\).
Focus: \(F(0, \pm \sqrt{15})\)
Vertices: \(V(0, \pm 4)\)
Major axis = \(8\)
Minor axis = \(2\)
Eccentricity: \(e = \dfrac{\sqrt{15}}{4}\)
Latus rectum = \(\dfrac{1}{2}\)
In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
For the ellipse \(4x^2 + 9y^2 = 36\).
Focus: \(F(\pm \sqrt{5}, 0)\)
Vertices: \(V(\pm 3, 0)\)
Major axis = \(6\)
Minor axis = \(4\)
Eccentricity: \(e = \dfrac{\sqrt{5}}{3}\)
Latus rectum = \(\dfrac{8}{3}\)
In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions.
Vertices \((\pm 5, 0)\) and foci \((\pm 4, 0)\).
\(\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1\)
In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions.
Vertices \((0, \pm 13)\) and foci \((0, \pm 5)\).
\(\dfrac{x^2}{144} + \dfrac{y^2}{169} = 1\)
In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions.
Vertices \((\pm 6, 0)\) and foci \((\pm 4, 0)\).
\(\dfrac{x^2}{36} + \dfrac{y^2}{20} = 1\)
In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions.
Ends of major axis \((\pm 3, 0)\) and ends of minor axis \((0, \pm 2)\).
\(\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1\)
In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions.
Ends of major axis \((0, \pm \sqrt{5})\) and ends of minor axis \((\pm 1, 0)\).
\(\dfrac{x^2}{1} + \dfrac{y^2}{5} = 1\)
In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions.
Length of major axis \(26\) and foci \((\pm 5, 0)\).
\(\dfrac{x^2}{169} + \dfrac{y^2}{144} = 1\)
In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions.
Length of minor axis \(16\) and foci \((0, \pm 6)\).
\(\dfrac{x^2}{64} + \dfrac{y^2}{100} = 1\)
In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions.
Foci \((\pm 3, 0)\) with \(a = 4\).
\(\dfrac{x^2}{16} + \dfrac{y^2}{7} = 1\)
In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions.
\(b = 3\), \(c = 4\), centre at the origin and foci on the x-axis.
\(\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1\)
In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions.
Centre at \((0, 0)\), major axis on the y-axis and passes through the points \((3, 2)\) and \((1, 6)\).
\(\dfrac{x^2}{10} + \dfrac{y^2}{40} = 1\)
In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions.
Major axis on the x-axis and passes through the points \((4, 3)\) and \((6, 2)\).
\(x^2 + 4y^2 = 52\) or \(\dfrac{x^2}{52} + \dfrac{y^2}{13} = 1\)
For the hyperbola \(\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1\), find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum.
Foci \((\pm 5, 0)\)
Vertices \((\pm 4, 0)\)
\(e = \dfrac{5}{4}\)
Latus rectum = \(\dfrac{9}{2}\)
For the hyperbola \(\dfrac{y^2}{9} - \dfrac{x^2}{27} = 1\), find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum.
Foci \((0, \pm 6)\)
Vertices \((0, \pm 3)\)
\(e = 2\)
Latus rectum = 18
For the hyperbola \(9y^2 - 4x^2 = 36\), find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum.
Foci \((0, \pm \sqrt{13})\)
Vertices \((0, \pm 2)\)
\(e = \dfrac{\sqrt{13}}{2}\)
Latus rectum = 9
For the hyperbola \(16x^2 - 9y^2 = 576\), find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum.
Foci \((\pm 10, 0)\)
Vertices \((\pm 6, 0)\)
\(e = \dfrac{5}{3}\)
Latus rectum = \(\dfrac{64}{3}\)
For the hyperbola \(5y^2 - 9x^2 = 36\), find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum.
Foci \(\left(0, \pm \dfrac{2\sqrt{14}}{\sqrt{5}}\right)\)
Vertices \(\left(0, \pm \dfrac{6}{\sqrt{5}}\right)\)
\(e = \dfrac{\sqrt{14}}{3}\)
Latus rectum = \(\dfrac{4\sqrt{5}}{3}\)
For the hyperbola \(49y^2 - 16x^2 = 784\), find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum.
Foci \((0, \pm \sqrt{65})\)
Vertices \((0, \pm 4)\)
\(e = \dfrac{\sqrt{65}}{4}\)
Latus rectum = \(\dfrac{49}{2}\)
Find the equation of the hyperbola with vertices \((\pm 2, 0)\) and foci \((\pm 3, 0)\).
\(\dfrac{x^2}{4} - \dfrac{y^2}{5} = 1\)
Find the equation of the hyperbola with vertices \((0, \pm 5)\) and foci \((0, \pm 8)\).
\(\dfrac{y^2}{25} - \dfrac{x^2}{39} = 1\)
Find the equation of the hyperbola with vertices \((0, \pm 3)\) and foci \((0, \pm 5)\).
\(\dfrac{y^2}{9} - \dfrac{x^2}{16} = 1\)
Find the equation of the hyperbola whose foci are \((\pm 5, 0)\) and whose transverse axis is of length 8.
\(\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1\)
Find the equation of the hyperbola whose foci are \((0, \pm 13)\) and whose conjugate axis is of length 24.
\(\dfrac{y^2}{25} - \dfrac{x^2}{144} = 1\)
Find the equation of the hyperbola whose foci are \((\pm 3\sqrt{5}, 0)\) and whose latus rectum is of length 8.
\(\dfrac{x^2}{25} - \dfrac{y^2}{20} = 1\)
Find the equation of the hyperbola whose foci are \((\pm 4, 0)\) and whose latus rectum is of length 12.
\(\dfrac{x^2}{4} - \dfrac{y^2}{12} = 1\)
Find the equation of the hyperbola whose vertices are \((\pm 7, 0)\) and whose eccentricity is \(e = \dfrac{4}{3}\).
\(\dfrac{x^2}{49} - \dfrac{9y^2}{343} = 1\)
Find the equation of the hyperbola whose foci are \((0, \pm \sqrt{10})\) and which passes through the point \((2, 3)\).
\(\dfrac{y^2}{5} - \dfrac{x^2}{5} = 1\)
If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
Focus is at the mid-point of the given diameter.
An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?
2.23 m (approx.)
The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.
9.11 m (approx.)
An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.
1.56 m (approx.)
A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.
\( \dfrac{x^2}{81} + \dfrac{y^2}{9} = 1 \)
Find the area of the triangle formed by the lines joining the vertex of the parabola \(x^2 = 12y\) to the ends of its latus rectum.
18 sq units
A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.
\( \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1 \)
An equilateral triangle is inscribed in the parabola \(y^2 = 4ax\), where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
\( 8\sqrt{3}a \)