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A circle is the locus of a point in a pl...

A circle is the locus of a point in a plane such that its distance from a fixed point in the plane is constant. Anologously, a sphere is the locus of a point in space such that its distance from a fixed point in space in constant. The fixed point is called the centre and the constant distance is called the radius of the circle/sphere. In anology with the equation of the circle `|z-c|=a`, the equation of a sphere of radius is `|r-c|=a`, where c is the position vector of the centre and r is the position vector of any point on the surface of the sphere. In Cartesian system, the equation of the sphere, with centre at `(-g, -f, -h)` is `x^2+y^2+z^2+2gx+2fy+2hz+c=0` and its radius is `sqrt(f^2+g^2+h^2-c)`. Q. Radius of the sphere, with `(2, -3, 4) and (-5, 6, -7)` as xtremities of a diameter, is

A

`sqrt((251)/(2))`

B

`sqrt((251)/(3))`

C

`sqrt((251)/(4))`

D

`sqrt((251)/(5))`

Text Solution

Verified by Experts

The correct Answer is:
C
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A circle is the locus of a point in a plane such that its distance from a fixed point in the plane is constant. Anologously, a sphere is the locus of a point in space such that its distance from a fixed point in space in constant. The fixed point is called the centre and the constant distance is called the radius of the circle/sphere. In anology with the equation of the circle |z-c|=a , the equation of a sphere of radius is |r-c|=a , where c is the position vector of the centre and r is the position vector of any point on the surface of the sphere. In Cartesian system, the equation of the sphere, with centre at (-g, -f, -h) is x^2+y^2+z^2+2gx+2fy+2hz+c=0 and its radius is sqrt(f^2+g^2+h^2-c) . Q. The centre of the sphere (x-4)(x+4)+(y-3)(y+3)+z^2=0 is

A circle is the locus of a point in a plane such that its distance from a fixed point in the plane is constant. Anologously, a sphere is the locus of a point in space such that its distance from a fixed point in space in constant. The fixed point is called the centre and the constant distance is called the radius of the circle/sphere. In anology with the equation of the circle |z-c|=a , the equation of a sphere of radius is |r-c|=a , where c is the position vector of the centre and r is the position vector of any point on the surface of the sphere. In Cartesian system, the equation of the sphere, with centre at (-g, -f, -h) is x^2+y^2+z^2+2gx+2fy+2hz+c=0 and its radius is sqrt(f^2+g^2+h^2-c) . Q. Equation of the sphere having centre at (3, 6, -4) and touching the plane rcdot(2hat(i)-2hat(j)-hat(k))=10 is (x-3)^2+(y-6)^2+(z+4)^2=k^4 , where k is equal to

Knowledge Check

  • The locus of a point moving in a space which is at a constant distance from a fixed point in space is called a ____

    A
    square
    B
    sphere
    C
    circle
    D
    triangle

  • The locus of a point which moves such that its distance from the point (0,0)is twice its distance from the Y axis is

    A
    `x^2-y^2=0`
    B
    `x^2-3y^2=0`
    C
    `3x^2-y^2=0`
    D
    none of the above

  • The locus of a point which moves such that its distance from the point (0,0)is twice its distance from the Y axis is

    A
    `x^2-y^2=0`
    B
    `x^2-3y^2=0`
    C
    `3x^2-y^2=0`
    D
    none of the above

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    The locus of a point which moves such that its distance from a given point is constant is

    Circle is the set of all those points in a plane whose distance from a fixed point remains constant.

    Sphere the set of all points in space which are equidistant from a fixed point is called a sphere.

    The locus of the point which moves such that the ratio of its distance from two fixed point in the plane is always a constant k(< 1) is

    Find the locus of a point which moves such that its distance from the origin is three xx its distance from x-axis.

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