Analytic Geometry-Plane, Sphere, Cone, Cylinder, Paraboloid, Ellipsoid, Hyperboloid [Optionals IAS Mains Mathematics]: Questions 1 - 8 of 25

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Question 1

Plane, Sphere, Cone, Cylinder, Paraboloid, Ellipsoid, Hyperboloid

Appeared in Year: 2003

Describe in Detail

Essay▾

Find the equations of the lines of intersection of the plane and the cone . (15 Marks)

Explanation

  • Here, equations of the plane and cone are,

  • Let the equation of the line of intersection of plan and the cone is given as,

  • Now, from question, this line (eq. 1) is on plane:

  • Also line of equation (1) is on given cone.

  • From equation (2) ,

  • Substitute the value of equation (4) in equation (3)

  • Dividing above equation by 35 and rearranged above equation, .

  • Usi…

… (21 more words) …

Question 2

Plane, Sphere, Cone, Cylinder, Paraboloid, Ellipsoid, Hyperboloid

Describe in Detail

Essay▾

Prove that the plane cuts the cone in perpendicular lines, if . (15 marks)

Explanation

  • Let the line of intersection of plane and cone is given as,

  • Equation (1) lies on plane and cone,

  • Substitute the value of equation (2) in equation (3) ,

  • Dividing above equation by both sides,

  • Equation (4) is a quadratic equation of .
  • If the roots of equation (4) are, ; then product of roots is

  • Therefore we can write,

  • Now, as we know, the two lines are per…

… (4 more words) …

Question 3

Plane, Sphere, Cone, Cylinder, Paraboloid, Ellipsoid, Hyperboloid

Appeared in Year: 2016

Describe in Detail

Essay▾

Find the equation of sphere which passes through the circle and is cut by the plane in a circle of radius [CS (Main) Paper 1]

Explanation

Equation of Sphere Which Passes through the Circle

The given circle is

or

Any sphere through these circle is

Its centre is and radius,

Now, sphere cut by the plane

In a circle of radius 3.

Draw CO plane from C.

Now

Equation of sphere is

Question 4

Plane, Sphere, Cone, Cylinder, Paraboloid, Ellipsoid, Hyperboloid

Appeared in Year: 2015

Describe in Detail

Essay▾

If represents one of the three mutually perpendicular generators of the cone then obtain the equations of the other two generators. [CS (Main) Paper 1]

Explanation

  • The given cone is

  • Here sum of the coefficient of is

    Cone has three mutually perpendicular generators.

  • The Equation of one of the three generator is

  • The equation of plane passing through the vertex of the cone and perpendicular to line is

  • Now the line of intersection of cone and the plane are the required lines.
  • So, we find these two lines.
  • Let th…

… (46 more words) …

Question 5

Plane, Sphere, Cone, Cylinder, Paraboloid, Ellipsoid, Hyperboloid

Appeared in Year: 2016

Describe in Detail

Essay▾

Show that the family of parabolas is self-orthogonal. [CS (Main) Paper 1]

Explanation

  • Given family of parabolas is

  • Differentiate eq. (1) w. r. t. x

  • Put the value of in , we get

  • Now for orthogonal replace
  • Then

  • Here equation and are same
  • Hence this trajectory is self-orthogonal.

Question 6

Plane, Sphere, Cone, Cylinder, Paraboloid, Ellipsoid, Hyperboloid

Appeared in Year: 2015

Describe in Detail

Essay▾

For what positive value of , the plane touches the sphere and hence find the point of contact. [CS (Main) Paper 1]

Explanation

  • The equation of sphere is

  • Its centre is

    and radius

  • The Equation of plane is

  • The plane will touch the sphere is perpendicular distance of centre from plane

    if

    if

    if

    if

    if

    if

    if

  • To find point of contact
  • The direction ratio՚s of the normal to the plane are
  • The equation of line through the centre of the sphere plane are

    Any point on it is if it lies …

… (10 more words) …

Question 7

Plane, Sphere, Cone, Cylinder, Paraboloid, Ellipsoid, Hyperboloid

Appeared in Year: 2015

Describe in Detail

Essay▾

Obtain the equation of the plane passing through the points and parallel to X axis. (CS Paper 1)

Explanation

The Equation of plane passing through is

This passes through the point

The plane (1) is parallel to the line

Therefore, normal to the plane is perpendicular to the line

Solving (2) and (3) we get

Substituting the value of in (1) , we get

This is the required equation of plane.

Question 8

Plane, Sphere, Cone, Cylinder, Paraboloid, Ellipsoid, Hyperboloid

Appeared in Year: 2015

Describe in Detail

Essay▾

Verify is the lines:

And

Are coplanar. If yes, then find the equation of the plane in which they lie. (CS Paper 1)

Explanation

We know that the lines

And are coplanar if

The equation of the given lines are

And

Here,

Applying

[Here are proportional]

Hence, given lines are coplanar.

The equation of plane containing the lines is

This is required equation of plane.