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

The equation of the normal to the circle x² + y² − 2x − 2y + 1 = 0 which is parallel to the line 2x + 4y = 3 is

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

If x + y = k is a normal to the parabola y² = 12x, then the value of k is

The equation of the circle passing through the foci of the ellipse  \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) having centre at (0, 3) is

Area of the greatest rectangle inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is

The eccentricity of the ellipse (x−3)² +(y−4)² \(=\frac { { y }^{ 2 } }{ 9 } \) is

The locus of a point whose distance from (-2,0) is \(\frac { 2 }{ 3 } \) times its distance from the line x = \(\frac { -9 }{ 2 } \) is

If the coordinates at one end of a diameter of the circle x² + y² − 8x − 4y + c = 0 are (11, 2), the coordinates of the other end are

(1) x = a cos θ, y = a sin θ
(2) θ
(3) 0 ≤ θ ≤ 2ㅠ
(4) (a cos θ, b sin θ)

(1) Transverse axis is parallel to x-axis
(2) Direction are x = ± \(\frac{a}{e}\)
(3) Centre is (0, 0)
(4) Transverse axis parallel to y-axis

(1) Vertex (h, k)
(2) Equation of directrix is x = h + a
(3) Axis of symmetry is y = k
(4) Length of latus rectum = 4a

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

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.

If y = 2\(\sqrt2\)x + c is a tangent to the circle x² + y² = 16, find the value of c.

Identify the type of conic section for each of the equations.
3x²+3y²−4x+3y+10 = 0

Identify the type of conic section for each of the equations.
x² + y² + x − y = 0

Identify the type of conic section for each of the equations.
y²+4x+3y+4 = 0

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

A line 3x+4y+10 = 0 cuts a chord of length 6 units on a circle with centre of the circle (2,1). Find the equation of the circle in general form.

Find the equation of circles that touch both the axes and pass through (-4, -2) in general form.

Find the equation of the tangent and normal to the circle x²+y²−6x+6y−8 = 0 at (2, 2) .

Find the equation of the parabola with focus \(\left( -\sqrt { 2 } ,0 \right) \) and directrix x =\(\sqrt { 2 } \).

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

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

 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

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.