REVISION ALL

TRIGONOMETRY 

Compound Angle 

• An angle made up of the algebraic sum of two or more angles is called compound angle 

i) sin(A+ B)= sinA cosB + cosA sinB

ii) sin(A- B)= sinA cosB - cosA sinB

iii) cos(A+ B)= cosA cosB - sinA sinB

iv) cos(A- B)= cosA cosB + sinA sinB

v) tan(A+ B)= (tanA + tanB)/(1 - tanA tanB)

vi) tan(45+ A)= (1 + tanA)/(1 - tanA)

vii) tan(A - B)= (tanA - tanB)/(1 + tanA tanB)

viii) tan(45- A)= (1 - tanA)/(1 + tanA)

ix) cot(A+ B)= (cotA cotB -1)/(cotA+ cotB)

x) cot(A- B)= (cotA cotB +1)/(cotA- cotB)

xi) sin(A+ B) sin(A - B)= sin²A - sin²B

                                      = Cos²B - cos²A

xii) cos(A+ B) cos(A - B)= cos²A - sin²B

                                      = Cos²B - sin²A

xiii) tan(A+ B + C)= (tanA + tanB+ tanC - tanA tanB tanC)/(1 - tanA tanB - tanB tanC - tanC tanA).

MULTIPLE AND SU MULTIPLE 

1) Sin2A= 2 sinA cosA = 2 tanA/(1+ tan²A)

2) cos2A= cos²A- sin²A = 1- 2 sin²A= 2 cos²A -1= (1- tan²A/(1+ tan²A).

3) 1+ cos2A= 2 cos²A , 1- cos2A= 2 sin²A

4) tan2A= 2tanA/(1- tan²A), where A≠ (2n +1)π/4

5) sin3A = 3SinA - 4sin²A

               = 4 sin(60°-A)sin(60° +A)

6) cos3A= 4cos³A - 3 cosA

               = 4 cos(60°-A) cosA cos(60°+A)

7)  tan3A= (3tanA - tan³A)/((1- 3tan²A)

                = tan(60° -A) tanA tan(60° +A)

PRODUCT 

1) 2 sinA cosB = sin(A + B) + sin(A - B)

2) 2 coA sinB = sin(A + B) - sin(A - B)

3) 2 cosA cosB = cos(A + B)+ cos(A - B)

4) 2 sinA sinB = cos(A - B) - cos(A + B)

SUM

1) SinC + sinD= 2 sin{(C+ D)/2} cos{(C - D)/2}

2) SinC - sinD= 2 cos{(C+ D)/2} sin{(C - D)/2}

3) cosC + cosD= 2 cos{(C+ D)/2} cos{(C - D)/2}

4) cosC  - cosD= - 2 sin{(C+ D)/2} sin{(C - D)/2}

5) tanA+ tanB= sin(A+ B)/(cosA cosB)

6) tanA- tanB= sin(A- B)/(cosA cosB) where A, B ≠ np + p/2

MAXIMUM AND MINIMUM VALUE OF 

f(0)= a cosθ + b sinθ, θ ∈ R

Let a= r sinα, b = r cosα so that r= √(a²+ b²), r ∈R

Also, a cosθ + b sinθ = r(cosθ sinα+ sinθ cosα)= r sin(θ +α)

Now the maximum and minimum values of sin(θ +α) are 1 and -1 respectively 

Hence - r≤ r sin(θ +α)≤ r

=> - √(a²+ b²) ≤ a cosθ + b sinθ ≤ √(a²+ b²)

Hence the maximum value is √(a²+ b²) and minimum value is - √(a²+ b²)

Conditional Identities 

When the angels A, B, C satisfy a given relation, many interesting identities can be established connecting the trigonometric functions of these angles, In providing these identities, we require the properties of complementary and supplementary angles. For example, if A+ B+ C=π, then

• sin(B+ C)= sinA, cosB = - cos(C+ A).

• Cos(A+ B)= - cosC, sinC= sin(A+ B)

• tan(C+ A)= - tanB, cotA = - cot(B+ C)

• cos{(A+B)/2= sin(C/2), cos(C/2)= = sin{A+B)/2}

• Sin{(C+ A)/2}= cos(B/2), sun(A/2) = cos{(B+C)/2}

• tan{B+ C)/2}= cot A, tan(B/2)= cot{(C+A)/2}

Some Important Identities

If A, B, C are angles of a triangle (or A+ B+ C)=π):

• tanA + tanB + tanC= tanA tanB tanC

• cotA + cotB + cotC= tanA tanB tanC

• tan(A/2)tan(B/2) + tan(B/2)tan(C/2) + tan(C/2) tan(A/2) = 1

• cot(A/2) + cot(B/2) + cot(C/2) = cot(C/2) cor(B/2) cot(C/2).

• sin2A + sin2B + sin2C= 4 sinA sinB sinC.

• cos2A cos2B + cos2C= -1 - 4 cosA cosB cosC.

• sinA + sinB + sinC= 4 cos(A/2) cos(B/2) cos(C/2).

• cosA + cosB + cosC= 1+ 4 sin(A/2) sin(B/2) sin(C/2).

Formula for General Solutions 

• tan²θ = tan²α = θ = nπ ± α 

Where α ∈ [i.π/2), n ∈ I

• sinθ = 0<=> θ= nπ,  n ∈ I

•  cosθ = 0 <=> θ= (2n +1)ⁿπ/2, n ∈ I

• tanθ = 0 <=> θ= nπ, n ∈ I

• sinθ = 1 <=> θ= (4n +1)π/2, n ∈ I

• sinθ = -1 <=> θ= (4n -1)π/2, n ∈ I

• cosθ = 1 <=> θ= 2nπ/2, n ∈ I

• cosθ = -1 <=> θ= (2n +1)π, n ∈ I

• sinθ = sin α => θ= nπ +(-1)ⁿα, α ∈ [-π/2, π/2]

• cosθ = cos α => θ= 2nπ ± α, α ∈ [0, π] n ∈ I

SALIENT POINTS 

* The general solution should be given unless the solution is required in a specified interval or range.

* While solving a trigonometric equation the equation at any steps should be avoided as far as possible. If squaring is necessary, check the solution for extraneous values.

* Never cancel terms containing unknown quantities on the two sides, which are in product. It may cause loss of genuine solution.

* The answer should not contain such values of angles, which make any of the terms undefined or infinite.

* Domain should not be changed. if it changed, necessary corrections must be incorporated.

* Check, that the denominator is not zero at any stage while solving equations.

* While solving trigonometric equations you may get same set of solution repeated in your answer. It is necessary for you to exclude these repetitions. eg., nπ+ π/2, forms a part of kπ/5 + π/10, k ∈ I the second part of second set of solution (you can check by putting k= 5m +2 (m∈ I).  Hence the final answer is kπ/5 + π/10 , k ∈ I.

* Sometimes the two solutions set consist partly of common values. In all such cases the common part must be represented only once.

Definition 

If f: A--> B is one to one and onto function and g is a rule under which for every element y ∈ B there exists and unique element x ∈ A then g : B --> A is called inverse function of f : A--> A, i.e., g= f⁻¹

So x= g(y) => x = f⁻¹(y)

So y= f(x) and x = g(y) such that then f(g(y))= y and x= g(f(x)) then f and g are said to be inverse function of each other.

* f⁻¹(x) ≠ [f(x)]⁻¹

* are sinx = sin⁻¹x

* arc sinx = 2nπ + arc sinx, n ∈ I

* Only one one onto function has an inverse function.

S.n   Function Domain  Range 

1.     sinx         x ∈ R     y ∈ [-1,1]

2.     cosx        x ∈ R     y ∈ [-1,1]

3. tanx   x ∈ R-(2n+1)π/2; n∈I  y ∈ R

4. Cotx  x ∈ R- nπ; n∈I      y ∈ R

5. cosecx   x ∈ R- nπ; n∈I   y ∈ (-∞,-1]U[1,∞)

6. secx   x ∈ R- (2n+1)π/2; n∈I y ∈ (-∞,-1]U[1,∞)

Domain, Range and Graph of Inverse Trigonometric functions 

where for the graph of take the plane mirror image of the portion of the graph on the line 

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[i.π/2), n ∈ I

= θ = nπ ± α 

 ∈

 tanA tanB tanC

 +α)α)

θ 

3- D vectors, plane etc

• Vector have magnitude and direction denoted by bar AB whereas scalars have only magnitude. The magnitude of vector is the length of the line segment AB denoted by |AB|

Types of Vectors 

• Zero vector: Initial point and terminal point coincide as bar AA

• Unit vector: magnitude is unity as cap a

• Coinitial vectors: Vectors having same initial point as bar OA, OC, OD

• Collinear vectors: Vectors which are parallel to the same vector.   .--<-.->--> .

• Equal vectors: vectors having some magnitude and same direction     bar a= bar b.

• Nagative of a vector: vector having some magnitude but opposite direction.  bar BA= - bar AB

• Free vectors: vectors whose initial point is not specified.

Laws of vectors 

a) Addition of vectors 

i) Triangle law of Addition: AC= AB+ BC

ii) Parallelogram law of Addition: OP+ OQ= OR

Properties of Addition of vectors 

i) a+ b = b+ a (cumulative)

ii) a+ (b + c)= (a+ b)+ c (Associative)

iii) a+ 0= a.       (Additive identity)

iv) a+ (-a)= 0 (additive inverse)

Multiplication of a vector by a scalar

• Let a be a vector and m a scalar, then ma is multiplication of vector a by scalar m.

|ma|= |m| a

Properties Multiplication of a vector by a scalar

i) m(-a)= -(ma)

ii) -m)(-a)= ma

iii) m(na)= (nm)a

iv) (m+ n)a= ma+ na

v) m(a+ b)= ma+ nb

• Two vectors a and b are colllinear or parallel iff a= mb for some nonzero scalar m.

• position vector:  Position vector of a point P(x,y,z) is given as bar OP= xi + yj + zk aits magnitude as |OP|= √(x²+ y²+ z²), where O is the origin.

Components of a vector in Two dimension

If a point P in a plane has coordinates (x,y) then 

i) OP= xi + yj

ii) |OP|= √(x²+ y²)

iii) The component of OP along x-axis is a vector xi, whose magnitude is |x| and whose direction is along OX or OX' according as x is positive or negative.

• Intercept form of the equation of a plane 

x/a + y/b + z/c= 1

• Equation of the plane passing through the intersection of two given planes

i) Vector form: r: (a₁ + λa₂) = d₁ + λd₂

ii) Cartesian form: (a₁x+ b₁y + c₁z - d₁) + λ(a₁x + b₁y + c₁z - d₂)= 0

• Coplanarity of two vectors 

i) Vector form: (a₂ - a₁). (b₁ x b₂)= 0

ii) Cartesian form:

|x₂ - x₁     y₂ - y₁      z₂ - z₁

    a₁           b₁            c₁     =0

    a₂           b₂            c₂

• Angle between two planes 

i) Vector form: cosθ= |(a₁. a₂)/+|n₁| |n₂|)|

ii) Cartesian form: 

cosθ=|(a₁a₂ + b₁b₂ + c₁c₂)/√(a₁²+ b₁²+ c₁²)√(a₂² + b₂²+ c₂²)|

• Distance of a plane from a plane 

i) Vector form: |(a.N - d)|/|N| , where N is normal to the plane.

ii) Cartesian form: |(ax₁+ by₁ + cz₁- d)/√(a²+ b² + c²)|

• Angle between a line and a plane 

i) Vector form:

φ = sin⁻¹|(b.n)/(|b| |n||

SAP- 1

Very short Type (1)

1) Find the equation of lines parallel to y-axis and passing through origin.    x/0= y/1= z/0

2) Show that the vector 3i + 5j + 2k, 2i - 3j - 5k and 5i+ 2j - 3k form the sides of an equilateral triangle.

3) Find the area of the parallelogram determined by the vector i + 2j + 3k and 3i - 2j + k.    8√3 square. units 

4) Direction ratioa of a line are 1,-2,3, find the direction cosines.     1.√14,-2/√14,3/√14

5) Write the vector equation of (x+5)/3 = (y -4)/2= (6- z)/2.      r= -5i+ 4j + 6k + λ(3i  + 2j - 3k)

Short Answer type (4)

6) Find the equation of plane containing the lines of intersection of the planes x+ y + z -6=0 and 2x + 3y + 4z +5=0 and passing through (1,1,1).        20x + 23y + 26z - 69=0

7) In ∆ OAB, E is the midpoint of OB and D is a point on AB such that AD: DB= 2:1. If OD AE intersect at P, determine the ratio OP: PD using vector methods.     3:2

8) Given three points whose position vectors are xi + yj + zk, i+ zj and - i - j. Find the condition for the point to be Collinear.        x - 2y = 1

9) If a,b,c are three non zero vectors such that a x b = c are b x c = a, show that a,b,c are mutually at right angles and |b|= 1 and |c|= |a|.

10) ABCD is a quadrilateral such that AB= b, AD= d, AC = mb + pd. Show that the area of the quadrilateral ABCD is (1/2) |m+ p| |b x d|.

11) Find the direction cosines of the two lines which are connected by the relations l - 5m + 3n=0 and 7l²+ 5m²- 3n²=0.     ±1.√14,-±2/√14,±3/√14 and ±1/√6, ±1/√6, ±2/√6

12) Show that the lines (x -1)/3= (y +1)/2 = (z -1)/5 and (x +2)/4 = (y -1)/3 = (z +1)/-2 do not intersect.

13) Show that the line x= ay + b, z= cy+ d and x= a'y + B', z= c'y + d' are perpendicular if aa' + cc' + 1=0

14) Find the angle between the lines 

x - 2y + z= 0= x + 2y - 2z and x + 2y + z= 0= 3x + 9y + 5z.       cos⁻¹(8/√406)

15) Find the equation of the plane through the points (1,0,-1), (3,2,2) and parallel to the line (x -1)/1 = (y +1)/-2 = (z -2)/3.       4x - y - 2z -6=0

Long Answer type (6)

16) Find the equation of the plane passing through the point (-1,2,1) and perpendicular to the line joining the points (-3,1,2) and (2,3,4). Find also the perpendicular distance of the origin from this plane.         r.(5i+ 2j + 2k)= 1,     1/√33

17) Find the shortest distance between the lines whose vector equation are

r= (i + 2j + 3k) + λ(2i + 3j + 4k) and r= (2i + 4j + 5k) + μ(4i + 6j + 8k)..    √5/√29

18) Find the distance of the point P(i + j + k) from the plane through the point A(2i + j + k), B(i + 2j + k) and C(i + j + 2k). Also, find the position vector of the foot of perpendicular from P on this plane.      1/√3, (4/3) (i + j + k)

19) Show (x +1)/3= (y +3)/5 = (z +5)/7 and (x -2)/1 = (y -4)/4 = (z -6)/7 are coplanar. Also, find the plane containing these two lines.

20) Find the image of the point (3,1,2) in the plane 2x - y + z= 4.        (1,2,1)

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MONOTONICITY 

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ROLLE'S THEOREM 

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Let Gaya the differences function for all prove that increase is increases find all the values of parameter for which the function increases no critical point prove that is differentiable on any Riyal there is an such that be continuous on and differentiable so that the existing such that prove that in equality is using determine which of the two numbers is a great time identify which is greater function suppose that on the interval 24 the function is the currency of 215 find the bounding function 24 using prove that all positive then show that show that for all interval which is increasing find the minimum value of differentiable function is double differential functions that prove that the exist some 33 such that 

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MAXIMUM AND MINIMUM 

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FORMULA

CONICS

1. Conic sections:

A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line.

a) The fixed point is called the focus.

b) The fixed straight line is called the directrix.

c) The constant ratio is called the eccentricity denoted by e.

d) The line passing through the focus & perpendicular to the directrix is called the axis.

e) A point of intersection of a conic with its axis is called vertex.

2. General equation of a conic: Focal Directrix Property:

The general equation of a conic with focus (p,q) & directrix lx + my + n=0 is

(l²+ m²)[(x - p)²+ (y - q)²]= e²(lx + my + n)² ≡ ax¹+ 2hxy+ by²+ 2gx + 2fy + c=0

3. Distinguish between the Conic:

The nature of the conic section depends upon the position of the focus S w.r.t. the directrix and also upon the value of the eccentricity e. Two different cases arise.

Case(i): when the focus lie on the directrix:

In this case D≡ abc+ 2fgh - af²- bg² - ch²=0 and the general equation of a conic represent a pair of straight lines and if:

e> 1 the line will be real and distinct intersecting at S.

e= 1 The lines will be coincident.

e < 1 The lines will be imaginary.

Case(ii): When the focus does not lie on the directrix:

e=1; D≠ 0 and h²= ab      a parabola 

0< e < 1; D≠ 0 and h²< ab  an ellipse 

D≠ 0; e> 1 and h²> ab.  a hyperbola 

e> 1; D≠ 0 and h²> ab; a+ b = 0 .   A rectangular hyperbola 

4. PARABOLA 

A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point (focus) is equal to its perpendicular distance from a fixed straight line (directrix).

Standard equation of a parabola is y²= 4ax. For this parabola:

i) Vertex is (0,0)

ii) Focus is (a,0)

iii) Axis is y=0

iv) Directrix is x+ a=0

a) Focal distance: 

The distance of a point on the parabola from the focus is called the focal distance of the point.

b) Focal chord:

A chord of the parabola, which passes through the focus is called a focal chord.

c) Double ordinate:

A chord of the parabola perpendicular to the axis of the symmetry is called a double ordinate.

d) Latus rectum:

A double ordinate passing through the focus of a focal chord perpendicular to the axis of parabola is called the latus rectum. For y²= 4ax.

• Length of the latus rectum= 4a.

• Length of the semi latus rectum= 2a.

• Ends of the latus rectum= L(a, 2a) & L'(a, -2a)

Notes that:

i) Perpendicular distance from focus on directrix= half the latus rectum.

ii) Vertex is middle point of the focus and the point of intersection of directrix and axis.

iii) Two parabola are said to be equal if they have the same latus rectum.

5. Parametric Representation:

The simplest and the best form of representing the coordinates of a point on the parabola is (at², 2at). The equation x= at² & y= 2at together represents the parabola y²= 4ax, t being the parameter.

6. Type of PARABOLA 

Four standards forms of the parabola are y²= 4ax; y²=- 4ax; x²= 4ay; x²= -4ay

When parabola y²= 4ax

Vertex:  (0,0)

Focus: (a,0)

Axis: y= 0

Directrix: x= - a

Length of latus rectum: 4a

Ends of Latus rectum: +a, ±2a)

Parametric equation: (at²,2at)

Focal length: x + a

When parabola y²= - 4ax

Vertex: (0,0)

Focus: (- a,0)

Axis: y= 0

Directrix: x= a

Length of latus rectum: 4a

Ends of Latus rectum: (-a, ±2a)

Parametric equation: (-at²,2at)

Focal length: x - a

When parabola x²= 4ay

Vertex:  (0,0)

Focus: (0,a)

Axis: x = 0

Directrix: y= - a

Length of latus rectum: 4a

Ends of Latus rectum: (±2a, a)

Parametric equation: (2at², at²)

Focal length: y + a

When parabola x²= - 4ay

Vertex: (0,0)

Focus: (0, - a)

Axis: x = 0

Directrix: y= a

Length of latus rectum: 4a

Ends of Latus rectum: (±2a, - a)

Parametric equation: (2at², - at²)

Focal length: y -  a

When parabola (y - k)²= 4a(x - h)

Vertex:  (h+ a, k)

Focus: (0,a)

Axis: y = k

Directrix: x+ a - h= 0

Length of latus rectum: 4a

Ends of Latus rectum: (h+ a, k± 2a)

Parametric equation: (h+ at², k+ 2at)

Focal length: x - h + a

When parabola (x - o)²= 4b(y - q)

Vertex: (p,q)

Focus: (p, b+ q)

Axis: x = p

Directrix: y+ b - q = 0

Length of latus rectum: 4apb

Ends of Latus rectum: (p± 2a, q+ a)

Parametric equation: (p+ 2at, q+ at²)

Focal length: y - q + b

7. Position of a point relative to a parabola:

The point (x₁, y₁) lies outside, on or inside the parabola y²= 4ax according as the expression y₁²- 4ax₁ is positive, zero or negative.

8. Chord joining two points:

The equation of a chord of the parabola y²= 4ax joining its two points P(t₁) and Q(t₂) is 

y(t₁ + t₂) = 2x + 2at₁t₂

Note:

i) If PQ is focal chord then t₁t₂= -1.

ii) Extremities of focal chord can be taken as (at², 2at) and (a/t², -2a/t).

9. LINE AND A PARABOLA :

a) The line y= mx + c meets the parabola y²= 4ax in two points real, coincident or imaginary according as > = < cm => condition of tangency is, c= a/m.

NOTE: Line y= mx + c will be tangent to parabrx²= 4ay if c= - am².

b) Length of the chord intercepted by the parabola y²= 4ax on the line y= mx + c is: (4/m²) √=a(1+ m¹)(a - mc)}.

NOTE 

Length of the focal chord making an angle α with x-axis is 4a coesec²α.

10. LENGTH OF SUBTANGENT & SUBNORMAL 

PT and PG are the tangent and normal respectively at the qP to the parabola y½= 4ax. Then 

TN= length of subtangent= twice the abscissa of the point P (Subtangent is always bisected by the vertex)

NG= length of subnormal which is constant for all points on the parabola and equal to its semilatus rectum (2a).

11. TANGENT TO THE PARABOLA y²= 4ax:

a) Point form:

Equation of tangent to the given parabola at its point (x₁ y₁) is 

yy₁ = 2a(x + x₁)

b) Slope form:

Equation of tangent to the given parabola whose slope is 'm', is 

y= mx + a/m, (m≠ 0)

Point of contact us (a/m², 2a/m)

c) Parametric form:

Equation of tangent to the given parabola at its point P(t), is ty= x + at²

NOTE:

Point of intersection of the tangents at the point t₁ & t₂ is [at₁t₂, a(t₁ + t₂)]

12. NORMAL TO THE PARABOLA y²= 4ax:

a) Point form:

Equation of normal to the given parabola at its point (x₁, y₁) is 

y- y₁ = - y₁(x - x₁)/2a.

b) Slope form:

Equation of normal to the given parabola whose slope is 'm', is 

y= mx - 2am - am³

foot of the normal is (am², - 2am)

c) Parametric form:

Equation of normal to the given parabola at its point P(t), is 

y+ tx = 2at + at²

NOTE 

i) Point of intersection of normals at t₁ & t₂ is [a(t₁²+ t₂²+ t₁t₂+2), - at₁t₂ (t₁ + t₂))

ii) If the normal to the parabola y²= 4ax at the point t₁, meets the parabola again at the point t₂ then t₂ = - (t₁ + 2/t₁).

iii) If normals to the parabola y²= 4ax at the point t₁ & t₂ intersect again on the parabola at the point  't₃'  then t₁ t₂ = 2; t₃ = -(r₁ + t₂) and the line joining t₁ and t₂ passes through a fixed point (-2a, 0).

i.e.,  am³+ m(2a - h)+ k =0

This gives m₁ + m₂+ m₃= 0; m₁ m₂+ m₂m₃+ m₃m₁= (2a - h)/a;  m₁m₂m₃= - k/a

Where m₁ , m₂, m₃ are the slopes of the three concurrent normas:

• Algebraic sum of slopes of the three concurrent normals is zero.

• Algebraic sum of ordinates of the three co-normal points on the parabola is zero.

• Centroid of the ∆ formed by three consecutive normal points lies on the axis of parabola (x-axis).

13. AN IMPORTANT CONCEPT:

If a family of straight lines can be represented by an equation λ²P+ λQ+ R= 0 where λ is a parameter and P,Q,R are linear functions of x and y then the family of lines will be tangent to the curve Q²= 4PR.

14. PAIR OF TANGENTS:

The equation of the pair of tangents which can be drawn from any point P(x₁ , x₂) outside the parabola to the parabola y²= 4ax is given by: SS₁ = T² where 

S= y²- 4ax; S₁= y₁² - 4ax₁; T ≡ yy₁ - 2a(x + x₁).

15. DIRECTOR CIRCLE:

Locus of the point of intersection of the perpendicular tangents to the parabola y²= 4ax is called the director circle. It's equation is x+ a = 0 which is parabola's own directrix.

16. CHORD OF CONTACT:

 Equation of the chord of contact of tangents drawn from a point P(x₁, y₁) is yy₁ = 2a(x + x₁)

NOTE

The area of the triangle formed by the tangents from the point (x₁, y₁) and the chord of contact us (y₁²- 4ax₁)³⁾²/2a i.e., (S₁)³⁾²/2a, also note that the chord of contact exists only if the point P is not inside.

17. CHORD WITH A GIVEN MIDDLE POINT 

Equation of the chord of the parabola y²= 4ax whose middle point is x₁, y₁ is 

y - y₁= 2a(x - x₁)/y₁.

This reduced to T= S₁ where T≡ 2a(x + x₁) and S₁ = y₁²- 4ax₁.

18 DIAMETER:

The locus of the middle point of a system of parallel chords of a parabola is called a DIAMETER. Equation to the diameter of a parabola is y= 2a/m, where m= slope of parallel chords.

19. IMPORTANT HIGHLIGHTS:

a) If the tangent and normal at any point P of the parabola intersect the axis at T and G then ST= SG= SP where S is the locus. In other words the tangent and the normal at a point P on the parabola are the bisectors of the angle between the focal radius SP and the perpendicular from P on the directrix. From this we conclude that all rays emanating from S will become parallel to the axis of the parabola after reflection.

b) The portion of a tangent to a parabola cut off between the directrix and the curve subtends a right angle at the focus.

c) The tangents at the extremities of a focal chord intersect at right angles on the directrix, and a circle on any focal chord as diameter touches the directrix. Also a circle on any focal radii of a point P(at², 2at) as diameter touches the tangents at the vertex and intercepts a chord of length a√(1+ t²) on a normal at the point P.

d) Any tangent to a parabola and the perpendicular on it from the focus meet on the tangent at the vertex.

e) Semi latus rectum of the parabola y²= 4ax, is the harmonic mean between segments of any focal chord of the parabola is; 2a = 2bc/(b + c) i.e., 1/b + 1/c = 1/a.

f) If the tangent at P and Q meet in T, then:

    i) TP and TQ subtend equal angles at the focus S.

   ii) ST²= SP. SQ

   iii) The triangles SPT and STQ are similar.

g) Tangents and Normals at the Extremities of the latus rectum of a parabola y²= 4ax constitue a square, their points of intersection being (-a,0) and (3a,0).

NOTE

i) The two tangents at the Extremities of focal chord meet on the foot of the directrix.

ii) Figure LNL'G is a square of side 2√2 a.

h) The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus.

ELLIPSE 

m₂, 

≡

ABC - MATHS