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Section of Solid

Conic Sections: Curves appear on the surface of a
cone when it is cut by some typical cutting planes

ELLIPSE θ >α ;θ <90o PARABOLA θ =α

θ <α
HYPERBOLA
θ ≥0

Section Plane Through Generators
Plane ⊥ axis. Circles ???

Section Plane Parallel to Axis.

Section Plane Parallel
to end generator.

Open/ unbounded

How do I identify ELLIPSE, PARABOLA, HYPERBOLA:

Locus of point moving in a plane such that the ratio of it’s distances

from a fixed point (focus) and a fixed line (directrix) always

remains

constant. A)

Ratio For

is called Ellipse

α ECCENTRICITY E<1

(E)

θ B) For Parabola E=1
C) For Hyperbola E>1

Assume

A

moving point

Line F

Fixed point

Line Fixed line

B
E = AF AB

V

PROBLEM:- Point F is 50 mm from a line AB. A point P is
moving in a plane such that the ratio of it’s distances from F and line AB remains constant and equals to 3/2. Draw locus of point P.

STEPS: 1 .Draw a vertical line AB and point F
50 mm from it. 2 .Divide 50 mm distance in 5 parts. 3 .Name 2nd part from F as V. It is 20mm
and 30mm from line AB and point F resp. It is first point giving ratio of it’s distances from AB and F 2/3 i.e 20/30 4 Form more points giving same ratio such as 30/45, 40/60, 50/75 etc. 5.Taking distances 30, 40 and 50mm from line AB, draw three vertical lines to the right side of it. 6. Now with 45, 60 and 75mm distances in compass cut these lines above and below, with F as center. 7. Join these points through V in smooth curve. This is required locus of P. It is an Hyberbola.

A
30mm (vertex)
V
B

HYPERBOLA DIRECTRIX FOCUS METHOD
F ( focus)

PROBLEM: Point F is 50 mm from a vertical straight line AB. Draw locus of point P, moving in a plane such that it always remains equidistant from point F and line AB.

PARABOLA
DIRECTRIX-FOCUS METHOD

SOLUTION STEPS: 1.Locate center of line (CF), perpendicular to AB. Bisect CF and find vertex V. 2.Mark 5 mm distance to right side of V, name those points 1,2,3,4 and from those draw lines parallel to AB. 4.Take C-1 distance as radius and F as center draw an arc cutting first parallel line to AB. Name upper point P1 and lower point P2. 5.Similarly repeat this process by taking again 5mm to right and locate P3P4. 6.Join all these points in smooth curve. It will be the locus of P equidistance from line AB and fixed point F.

A
C (VERTEX) V
1 2 3 4
B

F ( focus)

Ellipse

PROBLEM:- POINT F is 50 mm from a LINE AB. A POINT P is MOVING in a PLANE SUCH THAT

RATIO of IT’S DISTANCES (E) FROM F and LINE AB REMAINS CONSTANT and EQUALS TO 2/3.

DRAW LOCUS OF POINT P.

A STEPS:

1 .Draw a vertical line AB and point F

50 mm from it. 2 .Divide 50 mm distance in 5 parts.

45mm

3 .Name 2nd part from F as V. It is 20mm

and 30mm from F and AB line resp.

It is first point giving ratio of it’s

distances from F and AB 2/3 i.e 20/30 4 Form more points giving same ratio such
as 30/45, 40/60, 50/75 etc.

(vertex)
V

F ( focus)

5.Taking 45,60 and 75mm distances from

line AB, draw three vertical lines to the

right side of it.

6. Now with 30, 40 and 50mm distances in

P

compass cut these lines above and below,

with F as center.

7. Join these points through V in smooth

curve. This is required locus of P.

B
Portion of

Ellipse

Complete ELLIPSE:- Locus of a point (A) moving in a plane such that the SUM of it’s distances from TWO fixed points (FOCUS 1 & FOCUS 2) always remains constant. This sum equals to the length of major axis.
A3 + A4 = Length of major axis.

T1
Drawing a perpendicular to a line at a given point

• Draw the line AB

• With P as center and any convenient radius, draw an arc cutting AB at C (shown blue)

• With the same radius cut 2 equal divisions CD and DE (shown red)

• With same radius and centers D

and E, draw arcs (green and

brown) intersecting at Q

A

• PQ is the required perpendicular

Q

D

E

C

PB

8

T2
To draw a normal and a tangent to an arc or circle at a point P on it
ƒWith centre P and any convenient radius, mark off two arcs cutting the arc/circle at C and D.
ƒObtain QR, the perpendicular bisector of arc CD. QR is the required normal. ƒDraw the perpendicular ST to QR for the required tangent.
9

T3
Tangent to a given arc AB (or a circle) from a point P outside it.
ƒJoin the centre O with P and locate the midpoint M of OP. ƒWith M as a centre and radius = MO, mark an arc cutting the circle at Q. ƒJoin P with Q. PQ is the required tangent. ƒAnother tangent PQ’ can be drawn in a similar way.

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Section of Solid