Parabola Session 3

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Session
Parabola Session 3
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Session Objective

• Number of Normals Drawn From a Point
• Number of Tangents Drawn From a Point
• Director circle
• Equation of the Pair of Tangents
• Equation of Chord of Contact
• Equation of the Chord with middle point at (h, k)
• Diameter of the Parabola
• Parabola y ax2 bx c

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Number of Normals Drawn From a Point (h,k)
Parabola be y2 4ax let the slope of the normal
be m, then its equation is given by y mx 2am
am3 if it passes through (h,k) then k mh
2am am3 i.e. am3 (2a h)m k 0 This
shows from (h,k) there are three normals possible
(real/imaginary) as we get cubic in m
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Observations from am3 (2a h)m k 0

1. At least one of the normal is real as cubic
   equation have atleast one real root
2. The three feet of normals are called Co-Normal
   points given by (am12, 2am1 ), (am22, 2am2) and
   (am32, 2am3) where mis are the roots of the
   given cubic eqn
3. Sum of the ordinates of the co-normal points
   2a (m1 m2 m3) 0
4. Sum of slopes of normals at co-normal points 0
5. Centroid of triangle formed by co-normal points
   lies on axis of the parabola.

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Observations from am3 (2a h)m k 0
7. Thus we have following different cases arises

• 3 real and distinct roots m1, m2, m3 or m1, m2,
  m1m2
• 3 real in which 2 are equal m1, m2, m2 or 2m2,
  m2, m2
• 3 real, all equal m1, m1, m1 or 0, 0, 0 ? k 0 ,
  h 2a
• 1 real, 2 imaginary m1, ? ? i? (? ? 0)

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Number of Tangents Drawn From a Point (h,k)
Parabola be y2 4ax let the slope of the tangent
be m, then its equation is given by y mx a/m
if it passes through (h,k) then k mh
a/m i.e. hm2 km a 0 This shows from
(h,k) there are two tangents possible
(real/imaginary) as we get quadratic in m
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Observations from hm2 km a 0
Discriminant k2 4ah S1

1. S1 gt 0 Point is outside parabola 2 real
   distinct tangents
2. S1 0 Point is on the parabola Coincident
   tangents
3. S1 lt 0 Point is inside parabola No real
   tangent
4. m1 m2 k/h , m1m2 a/h

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Director Circle
Locus of the point of intersection of the
perpendicular tangents is called Director
Circle hm2 km a 0 m1m2 a/h 1 ? h a
i.e. locus is x a Hence in case of parabola
perpendicular tangents intersect at its
directrix. Director circle of a parabola is its
directrix.
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Equation of the Pair of Tangents
Parabola be y2 4ax then equation of pair of
tangents drawn from (h,k) is given by SS1
T2 where S ? y2 4ax, S1 ? k2 4ah and T ?
ky 2a(x h) Pair of Tangents (y2
4ax)(k2 4ah) (ky 2a(x h))2
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Equation of Chord of Contact
Parabola be y2 4ax then equation of chord of
contact of tangents drawn from (h,k) is given
by T 0 where T ? ky 2a(x h) Chord of
Contact is ky 2a(x h)
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Equation of the Chord with middle point at (h, k)
Parabola be y2 4ax then equation of chord whose
middle point is at (h,k) is given by T S1 where
T ? ky 2a(x h) and S1 ? k2 4ah Chord
with middle point at (h,k) is ky 2a(x h)
k2 4ah i.e. ky 2ax k2 2ah
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Diameter of the Parabola
Diameter Locus of mid point of a system of
parallel chords of a conic is known as diameter
Let (h, k) be the mid point of a chord of slop e
m then its equation is given by
ky 2ax k2 2ah if its slope is m then
Locus is
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Class Exercise - 1
Find the locus of the points from whichtwo of
the three normals coincides.
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Solution
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Solution contd..
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Class Exercise - 2
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Solution
m3 satisfies (i)
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Solution contd..
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Class Exercise - 3
Find the locus of the middle pointsof the normal
chords of the parabolay2 4ax.
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Solution
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Solution contd..
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Class Exercise - 4
Find the locus of the point ofintersection of
the tangents at theextremities of chord of y2
4ax whichsubtends right angle at its vertex.
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Solution
Let (h, k) be the point of intersection
of tangents then its chord of contact is given by
T 0 i.e. ky 2a (x h) 0 ...(i)
Now according to the question pair of
lines joining origin to the point of intersection
of (i) with the parabola are at right angles.
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Solution contd..
4ax2 2kxy hy2 0
Pair of lines are perpendicular if 4a h
0 hence locus of (h, k) is x 4a 0
Alternative
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Solution contd..
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Class Exercise - 5
Find the equation of the diameter of theparabola
given by 3y2 7x, whose systemof parallel
chords are y 2x c.
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Solution
Let (h, k) be the middle point of the chord then
its equation is given by T S1
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Solution contd..
Alternative
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Class Exercise - 6
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Solution
above equation have 3 real roots if 2 4c lt 0
Hence,answer is (c).
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Class Exercise - 7
The mid point of segmentintercepted by the
parabola x2 6yfrom the line x y 1 is ___.
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Solution
Let the mid point be (h, k), therefore,
its equation is given by hx 3 (y k) h2
6k or hx 3y h2 3k
Hence (3, 2) is the mid point of the segment.
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Class Exercise - 8
Draw y 2x2 3x 1.
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Solution
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Class Exercise - 9
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Solution
Let (h, k) be the point of intersection then
equation of pair of tangents are given by SS1 T2
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Solution contd..
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Class Exercise - 10
Find the coordinates of feet of thenormals drawn
from (14, 7) to theparabola y2 16x 8y.
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Solution
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Solution contd..