Comparing equation with y²=4ax.
4a=40. Length of latus rectum of parabola is 4a =40.

Comparing equation with x²=-4ay.
4a=100 => a=25. Line passing through focus perpendicular to axis is latus rectum.
Equation of latus rectum is y=-a => y=-25.

Comparing equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.
Conjugate axis length = 2b = 2*4 =8 units.

Comparing equation with \((\frac{x}{b})^2+(\frac{y}{a})^2\)=1, we get a=5 and b=4.
For ellipse, c²=a²-b²=25-16=9 => c=3.
So, coordinates of foci are (0,±c) i.e. (0,±3).

Equation of circle with center at (a, b) and radius r units is
(x-a)²+(y-b)²=r²
So, equation of circle is (x-2)²+(y-5)²=5² => x²+y²-4x-10y+4=0.

Equation of parabola which is symmetric about y-axis with vertex (0, 0) is x²=4ay
Since parabola pass through (6, 3) n 6²=4a*3 => a=3.
So, equation is x²=12y.

Circle has equation x²+y²-2x-4y=0.
0²+0²-2*0-4*0+0 = 0 so, point is on circle.

A hyperbola in which length of transverse and conjugate axis is equal is called equilateral hyperbola. In this type of hyperbola, a=b i.e. 2a=2b or length of transverse and conjugate axis are equal.

Axis is line passing through focus which divides parabola symmetrically into two equal halves. Equation of axis of parabola x²=24y is x=0.

Circle has equation x²+y²-2x-4y-16=0.
6²+2²-2*6-4*2-36 = 36+4-12-8-36 =-16<0 so, point is inside circle.