Properties of Conics
1
Parabola
P
In the following diagram, S is the focus, DD′ is the directrix, A the vertex, the line AS the axis and P an arbitrary point on the parabola. P T is the tangent and P G the normal to the parabola. N T is the sub tangent and N G is the sub normal. Also ZS = 2a is a fixed parameter of the parabola.
T B .. S
D P
M
Q
K
Y
T
Z
A
Figure 2:
G
.. N
S
↬ pp 11. Tangents drawn at the extremities of any chord intersect on the diameter bisecting that chord. (See Fig. 2) ↬ pp 12. If the tangents at the ends of a chord P Q intersect at T , then T P and T Q subtend equal angles at the focus; that is ∠P ST = ∠QST . Also, ∠P SQ = 2∠P T Q.
D′
Figure 1: ↬ pp 1 (Defn). P S = P M and P N = 4AS · AN 2
↬ pp 2. SP = ST = SG. Hence △P ST is isosceles and ∠SP T = ∠ST P . ↬ pp 3. The tangent P T bisects ∠SP M and hence M is the reflection of S on the tangent line P T .
↬ pp 13. If the tangents at the ends of a chord P Q intersect at T , then SP · SQ = ST 2 , S being the focus. ↬ pp 14. The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by tangents at these points. With reference to Fig. 3, Ar(△P QR) = 2 Ar(△BDE).
↬ pp 4. Mid-point of SM (which is Y in the diagram) lies on the tangent at the vertex.
P
↬ pp 5. The vertex can be obtained by drawing a line perpendicular to any tangent intersecting it at Y and then dropping a perpendicular on the axis from Y .
B
D
↬ pp 6. The vertex A bisects the sub tangent N T while the sub normal N G is a constant equal to half the latus rectum.
Q ..
S
E
↬ pp 7. If the tangent at P meets the directrix at K then ∠KSP = 90◦ because △KSP ∼ = △KM P . ↬ pp 8. Tangents drawn at the extremities of any focal chord intersect on the directrix at right angles.
R
↬ pp 9. The length of any focal chord inclined to the axis at an angle θ is 4a csc2 θ.
Figure 3:
For any conic, the diameter is the locus of the mid points of a family of parallel chords. For a parabola, a diameter is always a line parallel to the axis.
Note: If three points (at2i , 2ati ), i = 1, 2, 3 be taken on the standard parabola y 2 = 4ax, then the area of the triangle formed by these points is
↬ pp 10. Tangent drawn to the parabola at the point where a diameter meets it, is parallel to the chord which that diameter bisects. (See Fig. 2)
a2 |(t1 − t2 )(t2 − t3 )(t3 − t1 )|
Anant Kumar
while the area of the triangle formed by tangents at these Mob. No. 9002833857, 9932347531
Properties of Conics points is 1 2 a |(t1 − t2 )(t2 − t3 )(t3 − t1 )|. 2 ↬ pp 15. The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus. ↬ pp 16. The orthocentre of the triangle formed by any three tangents to a parabola, lie on the directrix. ↬ pp 17. If the normals at three points P , Q, and R meet in a point O, then SP · SQ · SR = AS · SO
2
where A and S are respectively the vertex and the focus of the parabola. ↬ pp 18. If the diameter bisecting chord AB intersects 3 the parabola at C, then area of △ACB = × area of 4 the parabolic segment ACB.
2
P T is the tangent and P G the normal at P . P M is the distance from the directrix. Also CA = CA′ = a is the semi-major axis of the ellipse, b its semi-minor axis, and e its eccentricity. The ∠φ shown is called the eccentric angle of the point P . b2 AN · A′ N . a2 ↬ pe 2. CS = CS ′ = ae and CZ = CZ ′ = a/e. ↬ pe 1 (Defn). P S = eP M , and P N 2 =
↬ pe 3. P S = a + eCN , P S ′ = a − eCN , so that P S + P S ′ = 2a = AA′ . PN b = , so that the ellipse can also be defined QN a as the locus of points which divide the semi-chords on a diameter of a circle in a fixed ratio. Also, CN = a cos φ and P N = b sin φ. ↬ pe 4.
↬ pe 5. CN · CT = a2 , CG = e2 CN , and SG = eSP . ↬ pe 6 (Reflection property). The normal SG bisects the angle SP S ′ internally while the tangent ST bisects this angle externally. Next refer to Figure 6. P Y is a tangent and P G the normal at an arbitrary point P of the ellipse.
B
Y
P Y′
A ..
E
C
.. A
S
C
G F
Figure 4:
Ellipse
S′
A′
g
Q P
M
Figure 6: K Z
A
S
.. C
φ
T G N
S
′
′
A
Z′
↬ pe 7. If SY and S ′ Y ′ be the perpendiculars from the foci upon the tangent at any point P of the ellipse, then Y and Y ′ lie on the auxiliary circle. ↬ pe 8. SY · S ′ Y ′ = b2 . Also CY ∥ S ′ P and CY ′ ∥ SP . ↬ pe 9. CY bisects SP and CY ′ bisects S ′ P .
Figure 5: In the diagram shown in Fig. 5, C is the center, S and S ′ are the focii, A and A′ the vertices of the ellipse. The line ZZ ′ is the axis while M Z and Z ′ K are the directrices. P is an arbitrary point on the ellipse and Q its corresponding point on the auxiliary circle. P N is the ordinate of P , Anant Kumar
↬ pe 10. The circle drawn on SP as diameter touches the auxiliary circle at Y . ↬ pe 11. If the normal at any point P meet the major and minor axes in G and g, and if CF be the perpendicular upon this normal, then P F · P G = b2 and P F · P g = a2 . Also P G · P g = SP · S ′ P , and CG · CT = CS 2 , where T is the point where the tangent at P intersects the axis of the ellipse. Mob. No. 9002833857, 9932347531
Properties of Conics
3
↬ pe 12. The tangent at the extremity of any diameter is parallel to the chords which it bisects. ↬ pe 13. The tangents at the ends of any chord meet on the diameter bisecting that chord. ↬ pe 14. If P CP ′ and DCD′ are a pair of conjugate diameters, the eccentric angles of P and D differ by a right angle.
K
P D N
A
.. C
S
A′
S′
L D′ P′ M
Figure 7: ↬ pe 15. CP 2 + CD2 = a2 + b2 ↬ pe 16. Area of parallelogram KLM N formed by tangents at the extremities of conjugate diameters is = 4ab. ↬ pe 17. SP · SP ′ = CD2 ↬ pe 18. If a pair of conjugate diameters are extended to meet a directrix, the orthocentre of the triangle thus formed lies at the corresponding focus.
Anant Kumar
Mob. No. 9002833857, 9932347531
