The radius of the circle passing through the point(6, 2) two of whose diameter are x + y = 6 and x + 2y = 4 is

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

Equation of tangent at (-4, -4) on x² = -4y is _____________

y² - 2x - 2y + 5 = 0 is a _________

The length of major and minor axes of 4x² + 3y² = 12 are ____________

The tangent at any point P on the ellipse \(\frac { { x }^{ 2 } }{ 6 } +\frac { { y }^{ 2 } }{ 3 } \) = 1 whose centre C meets the major axis at T and PN is the perpendicular to the major axis; The CN CT = ______________

(1) y² = 4ax
(2) c = \(\frac{a}{m}\)
(3) c² = a²(1+m²)
(4) \(\left( \frac { a }{ { m }^{ 2 } } ,\frac { 2a }{ m } \right) \)

Find the general equation of the circle whose diameter is the line segment joining the points (−4, −2)and (1, 1) is x²+y²+5x+3y+6=0

Examine the position of the point (2, 3) with respect to the circle x² + y² − 6x − 8y + 12 = 0.

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

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

Find the eccentricity of the ellipse with foci on x-axis if its latus rectum be equal to one half of its major axis.

A circle of radius 3 units touches both the axes. Find the equations of all possible circles formed in the general form.

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

Find the value of c if y = x + c is a tangent to the hyperbola 9x² - 16y² = 144.

Show that the line x + y + 1 = 0 touches the hyperbola \(\frac { { x }^{ 2 } }{ 16 } -\frac { { y }^{ 2 } }{ 15 } \) = 1 and find the co-ordinates of the point of contact

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.

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.

The foci of a hyperbola coincides with the foci of the ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { y^{ 2 } }{ 9 } =1\). Find the equation of the hyperbola if its eccentricity is 2.

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