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.

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

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 equation y = \(\frac { 1 }{ 32 } \)x² models cross sections of parabolic mirrors that are used for solar energy. There is a heating tube located at the focus of each parabola; how high is this tube located above the vertex of the parabola?

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?

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 :
\(\frac { { \left( x+3 \right) }^{ 2 } }{ 225 } -\frac { { \left( y-4 \right) }^{ 2 } }{ 64 } =1\)