The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x² + y² − 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to

The radius of the circle 3x² + by² + 4bx − 6by + b² = 0 is

If P(x, y) be any point on 16x² + 25y² = 400 with foci F₁ (3, 0) and F₂ (-3, 0) then PF₁  + PF₂ is

The ellipse \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E₂ passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse is

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

Obtain the equation of the circle for which (3, 4) and (2, -7) are the ends of a diameter.

Identify the type of the conic for the following equations :
11x²−25y²−44x+50y−256 = 0

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

Identify the type of the conic for the following equations:
(1) 16y² = −4x²+64
(2) x²+y² = −4x−y+4
(3) x²−2y = x+3
(4) 4x²−9y²−16x+18y−29 = 0

The parabolic communication antenna has a focus at 2m distance from the vertex of the antenna. Find the width of the antenna 3m from the vertex.

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.

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