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

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 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) .

Determine whether the points (-2, 1), (0, 0) and (-4, -3) lie outside, on or inside the circle x²+y²−5x+2y−5 = 0 .

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 parabola y² = 4ax.

The orbit of Halley’s Comet is an ellipse 36.18 astronomical units long and by 9.12 astronomical units wide. Find its eccentricity.

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.