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Featured 3 months ago

**Right Choice.**

Let, **radius** of the **circle** in **Question.**

Because the circle **touches** the **X-Axis** at

**centre C** must be,

Given that the point

This means that, the centre is

Hence, the eqns. of the circles are,

Of all the given points, only **satisfies** the above eqn.

Hence, **Right Choice.**

Featured 2 months ago

Using distance formula,

Featured 2 months ago

Given

Let

given

and

In

Or if you are not sure if

Featured 1 month ago

**(d) None of the Above.**

Clearly, the eqn. of the given line

In other words, **any point on**

Also, the

Hence, the right choice is **(d) None of the Above.**

Featured 1 month ago

Continuation.

The common-volume surface (CVS) is segmented by

equal segments, each reaching the common chord ( axis of

symmetry ) through planar sides, like segments of a peeled orange.

To get a typical segment, rotate the LHS circle in the planar-section

graph ( in my 1st part answer ) about the chord, through

rad. The RHS part of the circle generates a segment for the inner

surface, and correspondingly, the larger LHS part forms the

opposite segment, for the outer surface.

Let us study the limit, as

shaped ) prolate spheroid of semi-axes

a = b = 1/2 and c =

The outer surface

axes a = b = 1.5 and c = 1, with conical dimples at the poles that

are

The volume enclosed, in the limit, is nearly

Now, the elusive volume of the common-to

spheres is expressed as a double integral

V =

with limits

I had used cylindrical polar coordinates

referred to the common chord as z-axis and its center as origin.

Choosing the center of the LHS sphere as origin, this becomes

This is indeed a Gordian knot. So, I look for another method that

leads to a closed form solution. The planar section z = 0 for 8 ( N =

3 ) spheres appears below. The graph reveals most of the aspects

in the description.

graph{((x-0.5)^2+y^2-1)((x+0.5)^2+y^2-1)((y-0.5)^2+x^2-1)((y+0.5)^2+x^2-1)((x-0.3536)^2+(y-0.3536)^2-1)((x+0.3536)^2+(y-0.3536)^2-1)((y+0.3536)^2+(x+0.3536)^2-1)((y+0.3536)^2+(x-0.3536)^2-1)=0[-3 3 -1.5 1.5]}

Now, the common space has just disappeared at z =

graph{((x-0.5)^2+y^2-.25)((x+0.5)^2+y^2-.25)((y-0.5)^2+x^2-.25)((y+0.5)^2+x^2-.25)((x-0.3536)^2+(y-0.3536)^2-.25)((x+0.3536)^2+(y-0.3536)^2-.25)((y+0.3536)^2+(x+0.3536)^2-.25)((y+0.3536)^2+(x-0.3536)^2-.25)=0[-3 3 -1.5 1.5]}

The graphs are on uniform scale.

Featured 1 month ago

Hence, the orthocentre of

Let ,

We take,

So,

It is clear that,

Hence, **hypotenuse.**

**right angled triangle.**

Hence, the orthocentre of

Please see the graph:

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