Find the equation of the circle passing through the points (1, 1 ), (2, -1 ) and (3, 2) .

 A road bridge over an irrigation canal have two semi circular vents each with a span of 20m and the supporting pillars of width 2m. Use Figure to write the equations that represents the semi-verticular vents

Find the equation of the circle through the points (1, 0),(-1, 0) , and (0, 1) 

Determine whether the points (-2, 1), (0, 0) and (-4, -3) lie outside, on or inside the circle x²+y²−5x+2y−5 = 0 .

If the equation 3x²+(3−p)xy+qy²−2px = 8pq represents a circle, find p and q. Also determine the centre and radius of the circle

Find the equation of the ellipse whose eccentricity is \(\frac { 1 }{ 2 } \), one of the foci is(2, 3) and a directrix is x = 7. Also find the length of the major and minor axes of the ellipse.

Find the foci, vertices and length of major and minor axis of the conic 4x² + 36y² + 40x − 288y + 532 = 0 

For the ellipse 4x² + y² + 24x − 2y + 21 = 0, find the centre, vertices and the foci. Also prove that the length of latus rectum is 2  

Find the centre, foci, and eccentricity of the hyperbola 11x² − 25y² −44x + 50y −256 = 0

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following:
\(\frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 9 } =1\)

Prove that the length of the latus rectum of the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 is \(\frac { { 2b }^{ 2 } }{ a } \).

Show that the absolute value of difference of the focal distances of any point P on the hyperbola is the length of its transverse axis.

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
\(\frac { { \left( x-3 \right) }^{ 2 } }{ 225 } +\frac { { \left( y-4 \right) }^{ 2 } }{ 289 } =1\)

Find the equations of tangent and normal to the parabola x²+6x+4y+5 = 0 at (1, -3) .

Find the equations of tangent and normal to the ellipse x²+4y² = 32 when \(\theta =\frac { \pi }{ 4 } \)

Find the equations of the two tangents that can be drawn from (5, 2) to the ellipse 2x²+7y² = 14 .

Find the equations of tangents to the hyperbola \(\frac { { x }^{ 2 } }{ 16 } -\frac { { y }^{ 2 } }{ 64 } \) = 1 which are parallel to10x − 3y + 9 = 0.

Show that the line x−y+4 = 0 is a tangent to the ellipse x²+3y² = 12 . Also find the coordinates of the point of contact.

Find the equation of the tangent to the parabola y² = 16x perpendicular to 2x + 2y + 3 = 0.

Find the equation of the tangent at t = 2 to the parabola y² = 8x. (Hint: use parametric form)

Prove that the point of intersection of the tangents at ‘t₁’ and ‘t₂’ on the parabola y² = 4ax is \(\left[ at_{ 1 }t_{ 2 },a({ t }_{ 1 }+{ t }_{ 2 }) \right] .\)

If the normal at the point ‘t₁’ on the parabola y² = 4ax meets the parabola again at the point ‘t₂’, then prove that t₂ = -\(\left( { t }_{ 1 } + \frac { 2 }{ { t }_{ 1 } } \right) \)

A semielliptical archway over a one-way road has a height of 3m and a width of 12m. The truck has a width of 3m and a height of 2.7m. Will the truck clear the opening of the archway?

The maximum and minimum distances of the Earth from the Sun respectively are 152 × 10⁶ km and 94.5 × 10⁶ km. The Sun is at one focus of the elliptical orbit. Find the distance from the Sun to the other focus.

A search light has a parabolic reflector (has a cross-section that forms a ‘bowl’). The parabolic bowl is 40 cm wide from rim to rim and 30 cm deep. The bulb is located at the focus.
(1) What is the equation of the parabola used for reflector?
(2) How far from the vertex is the bulb to be placed so that the maximum distance covered?

A room 34m long is constructed to be a whispering gallery. The room has an elliptical ceiling, as shown in Figure. If the maximum height of the ceiling is 8 m, determine where the foci are located.

Two coast guard stations are located 600 km apart at points A(0, 0) and B(0, 600). A distress signal from a ship at P is received at slightly different times by two stations. It is determined that the ship is 200 km farther from station A than it is from station B. Determine the equation of hyperbola that passes through the location of the ship.

A bridge has a parabolic arch that is 10 m high in the centre and 30 m wide at the bottom. Find the height of the arch 6 m from the centre, on either sides.

A tunnel through a mountain for a four lane highway is to have a elliptical opening. The total width of the highway (not the opening) is to be 16 m, and the height at the edge of the road must be sufficient for a truck 4 m high to clear if the highest point of the opening is to be 5 m approximately. How wide must the opening be?

At a water fountain, water attains a maximum height of 4 m at horizontal distance of 0.5 m from its origin. If the path of water is a parabola, find the height of water at a horizontal distance of 0.75 m from the point of origin.

An engineer designs a satellite dish with a parabolic cross section. The dish is 5 m wide at the opening, and the focus is placed 1.2 m from the vertex
(a) Position a coordinate system with the origin at the vertex and the x -axis on the parabola’s axis of symmetry and find an equation of the parabola.
(b) Find the depth of the satellite dish at the vertex.

Parabolic cable of a 60m portion of the roadbed of a suspension bridge are positioned as shown below. Vertical Cables are to be spaced every 6m along this portion of the roadbed. Calculate the lengths of first two of these vertical cables from the vertex.

Cross section of a Nuclear cooling tower is in the shape of a hyperbola with equation\(\frac { { x }^{ 2 } }{ { 30 }^{ 2 } } -\frac { { y }^{ 2 } }{ { 44 }^{ 2 } } =1\). The tower is 150 m tall and the distance from the top of the tower to the centre of the hyperbola is half the distance from the base of the tower to the centre of the hyperbola. Find the diameter of the top and base of the tower.
 

A rod of length 1.2 m moves with its ends always touching the coordinate axes. The locus of a point P on the rod, which is 0.3 m from the end in contact with x -axis is an ellipse. Find the eccentricity.

Assume that water issuing from the end of a horizontal pipe, 7. 5 m above the ground, describes a parabolic path. The vertex of the parabolic path is at the end of the pipe. At a position 2. 5 m below the line of the pipe, the flow of water has curved outward 3m beyond the vertical line through the end of the pipe. How far beyond this vertical line will the water strike the ground?

On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4 m when it is 6 m away from the point of projection. Finally it reaches the ground 12 m away from the starting point. Find the angle of projection.

Points A and B are 10 km apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is 6 km closer to A than B. Show that the location of the explosion is restricted to a particular curve and find an equation of it.

Find the vertex, focus, equation of directrix and length of the latus rectum of the following:
x²−2x+8y+17= 0

Find the vertex, focus, equation of directrix and length of the latus rectum of the following: y²−4y−8x+12 = 0

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following:
\(\frac { { x }^{ 2 } }{ 3 } +\frac { { y }^{ 2 } }{ 10 } =1\)

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following:
\(\frac { { x }^{ 2 } }{ 25 } -\frac { { y }^{ 2 } }{ 144 } =1\)

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following:
\(\frac { { y }^{ 2 } }{ 16 } -\frac { { x }^{ 2 } }{ 9 } =1\)

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
\(\frac { { \left( x+1 \right) }^{ 2 } }{ 100 } +\frac { { \left( y-2 \right) }^{ 2 } }{ 64 } =1\)

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
\(\frac { { \left( x+3 \right) }^{ 2 } }{ 225 } -\frac { { \left( y-4 \right) }^{ 2 } }{ 64 } =1\)

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
\(\frac { { \left( y-2 \right) }^{ 2 } }{ 25 } \frac { { \left( x+1 \right) }^{ 2 } }{ 16 } =1\)

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
18x²+12y²−144x+48y+120 = 0

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
9x²−y²−36x−6y+18 = 0