The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is

The circle x² + y² = 4x + 8y +5 intersects the line 3x−4y = m at two distinct points if

The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { and } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\)

Tangents are drawn to the hyperbola  \(\frac { { x }^{ 2 } }{ 9 } -\frac { { y }^{ 2 } }{ 4 } =1\) parallel to the straight line 2x − y = 1. One of the points of contact of tangents on the hyperbola is

An ellipse has OB as semi minor axes, F and F′ its foci and the angle FBF′ is a right angle. Then the eccentricity of the ellipse is

y² - 2x - 2y + 5 = 0 is a _________

If a parabolic reflector is 20 cm in diameter and 5 cm in diameter and 5 cm deep, then its focus is ____________

The equation 7x²- 6\(\sqrt { 3 } \) xy + 13y² - 4\(\sqrt { 3 } \) x - 4y - 12 = 0 represents ____________

The distance between the foci of a hyperbola is 16 and e = \(\sqrt { 2 } \). Its equation is ____________

If the foci of the ellipse \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 and the hyperbola \(\frac { { x }^{ 2 } }{ 144 } -\frac { { y }^{ 2 } }{ 81 } =\frac { 1 }{ 25 } \) coincide then b² is __________

If e₁, e₂ are eccentricities of the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 and the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 then 

The equation of the director circle of the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 is ___________

The number of normals to the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 from an external point is ________

If t₁ and t₂ are the extremities of any focal chord of y² = 4ax then t₁t₂ is ______________

The number of normals that can be drawn from a point to the parabola y² = 4ax is __________

Find the general equation of a circle with centre (-3, -4) and radius 3 units.

If y = 4x + c is a tangent to the circle x² + y² = 9, find c 

Find the equation of the circle with centre (2, -1) and passing through the point (3, 6) in standard form.

Find centre and radius of the following circles.
 x²+ (y + 2)² = 0

Find centre and radius of the following circles.
2x²+2y²−6x+4y+2 = 0

Find the equation of tangent to the circle x² +y² + 2x - 3y - 8 = 0 at (2, 3).

If the line y = 3x + 1, touches the parabola y² = 4ax, find the length of the latus rectum?

For the ellipse x² + 3y² = a², find the length of major and minor axis.

Find the eccentricity of the hyperbola with foci on the x-axis if the length of its conjugate axis is \({ \left( \frac { 3 }{ 4 } \right) }^{ th }\) of the length of its tranverse axis.

Find the equation of the hyperbola whose vertices are (0, ±7) and e = \(\frac { 4 }{ 3 } \)

Find the equation of the circle described on the chord 3x + y + 5 = 0 of the circle x² + y² = 16 as diameter.

Find the centre and radius of the circle 3x² + (a + 1)y² + 6x − 9y + a + 4 = 0.

Find the equation of the circle with centre (2, 3) and passing through the intersection of the lines 3x − 2y − 1 = 0 and 4x + y − 27 = 0.

A circle of area 9π square units has two of its diameters along the lines x + y = 5 and x−y = 1. Find the equation of the circle.

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 length of Latus rectum of the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\)  

Find the equation of the hyperbola with vertices (0, ±4) and foci(0, ±6).

Find the equation of the ellipse in each of the cases given below:
length of latus rectum 4, distance between foci 4 \( \sqrt{ 2}\) , centre (0, 0) and major axis as y - axis.

Find the vertex, focus, equation of directrix and length of the latus rectum of the following:
y² = −8x

Find the circumference and area of the circle x² +y² - 2x + 5y + 7 = 0

 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

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

Certain telescopes contain both parabolic mirror and a hyperbolic mirror. In the telescope shown in figure the parabola and hyperbola share focus F₁ which is 14m above the vertex of the parabola. The hyperbola’s second focus F₂ is 2m above the parabola’s vertex. The vertex of the hyperbolic mirror is 1m below F₁. Position a coordinate system with the origin at the centre of the hyperbola and with the foci on the y-axis. Then find the equation of the hyperbola.

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.

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?

A kho-kho player In a practice session while running realises that the sum of tne distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him of the distance between the poles is 6m.