Use these Inter 2nd Year Maths 2B Formulas PDF Chapter 3 Parabola to solve questions creatively.

## Intermediate 2nd Year Maths 2B Parabola Formulas

**Definition:**

→ Conic section: If right circular solid cone is cut by a plane the section of it is called a Conic section.

→ Conic: The locus of a point whose distances from a fixed point and a fixed straight line are in constant ratio ‘e’ is called a conic.

→ Parabola : A conic with eccentricity 7 is called a parabola i.e., the locus of a point, whose distance from fixed point (focus) is equal to the distance from a fixed straight line (directrix) is called a parabola.

→ Axis of Parabola: The line passing through the vertex and the focus and perpendicular to directrix of the parabola is called the axis of the Parabola.

→ Equation of Parabola : General form – Let S(α, β) be the focus and ax + by + c = 0 be directrix then by definition the equation of parabola be,

\(\sqrt{(x-\alpha)^{2}+(y-\beta)^{2}}\) = \(\left|\frac{a x+b y+c}{\sqrt{a^{2}+b^{2}}}\right|\)

**Various forms of parabola:**

→ y^{2} = 4ax

Axis : X – axis

Focus : (a, 0)

Vertex : (0, 0)

Equation of directrix : x + a = 0

→ y^{2}= – 4ax

Axis: x – axis

Focus: (- a, 0)

Vertex : (0, 0)

Equation of directrix : x – a = 0

→ x^{2} = 4ay (a > 0)

Axis: Y-axis Focus : (0, a)

Vertex : (0, 0)

Equation of directrix: y + a = 0

→ x^{2} = -4ay(a>0)

Axis : Y axis

Focus : (0, – a)

Vertex: (0, 0)

Equation of directrix : y – a = 0

→ (y – k)^{2} = 4a (x – h) a > 0

Axis : y = k’ a line parallel to X- axis

Focus : (a + h, k)

Vertex: (h, k)

Equation of directrix: x + a = h

→ (x – h)^{2} = 4a (y – k) a > 0

Axis: x = h a line parallel to Y-axis

Focus: (h, a + k)

Vertex: (h, k)

Equation of directrix: y + a = k

→ \(\sqrt{(x-h)^{2}+(y-k)^{2}}\) = \(\left|\frac{a x+b y+c}{\sqrt{a^{2}+b^{2}}}\right|\)

Axis: b(x – h) – a(y – k) = 0

Focus: (h, k)

Vertex: Some point A(fig)

Equation of directrix : ax + by + c = 0

Chord: Line segment joining two points of a parabola is called a, chord of the parabola. If this chord passes through focus is called focal chord.

Chord passing through focus and ⊥ to axis is called latus rectum.

Length of latus rectum = 4a

Parametric form of parabola:

For y^{2} = 4ax

Parametric form will be x = at^{2}, y = 2at

Definition:

→ The locus of a point which moves in a plane so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line is called a conic section or conic. The fixed point is called focus, the fixed straight line is called directrix and the constant ratio ‘e’ is called eccentricity of the conic.

If e = 1, then the conic is called a Parabola.

If e < 1, then the conic is called an Ellipse. If e > 1, then the conic is called a hyperbola.

Note: The equation of a conic is of the form ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0.

→ Directrix of the Conic: A line L = 0 passing through the focus of a conic is said to be the principal axis of the conic if it is perpendicular to the directrix of the conic.

→ Vertices: The points of intersection of a conic and its principal axis are called vertices of the conic.

→ Centre: The midpoint o the line segment joining the vertices of a conic is called centre of the conic.

→ Note 1: If a conic has only one vertex then its centre coincides with the vertex.

→ Note 2: If a conic has two vertices then its centre does not coincide either of the vertices. In this case the conic is called a central conic.

→ Standard Form: A conic is said to be in the standard form if the principal axis of the conic is x-axis and the centre of the conic is the origin.

Equation of a Parabola in Standard Form:

The equation of a parabola in the standard form is y^{2} = 4ax.

Proof :

Let S be the focus and L = 0 be the directrix of the parabola.

Let P be a point on the parabola.

Let M, Z be the projections of P, S on the directrix L = 0 respectively.

Let N be the projection of P on SZ.

Let A be the midpoint of SZ.

Therefore, SA = AZ, ⇒ A lies on the parabola. Let AS = a.

Let AS, the principal axis of the parabola as x-axis and Ay perpendicular to SZ as y-axis.

Then S = (a, 0) and the parabola is in the standard form.

Let P = (x_{1}, y_{1}).

Now PM = NZ = NA + AZ = x_{1} + a

P lies on the parabola ⇒ \(\frac{\mathrm{PS}}{\mathrm{PM}}\) = 1 ⇒ PS = PM

⇒ \(\sqrt{\left(x_{1}-a\right)^{2}+\left(y_{1}-0\right)^{2}}\) = x_{1} + a

⇒ (x_{1} – a)^{2} + y_{1}^{2} = (x_{1} + a)^{2}

⇒ y_{1}^{2} = (x_{1} + a)^{2} – (x_{1} – a)^{2}

⇒ y_{1}^{2} = 4ax_{1}

The locus of P is y^{2} = 4ax.

∴ The equation to the parabola is y^{2} = 4ax.

**Nature of the Curve y ^{2} = 4ax.**

(i) The curve is symmetric with respect to the x-axis.

∴ The principal axis (x-axis) is an axis of the parabola.

(ii) y = 0 ⇒ x = 0. Thus the curve meets x-axis at only one point (0, 0).

Hence the parabola has only one vertex.

(iii) If x < 0 then there exists no y ∈ R. Thus the parabola does not lie in the second and third quadrants. (iv) If x > 0 then y^{2} > 0 and hence y has two real values (positive and negative). Thus the parabola lies in the first and fourth quadrants.

(v) x = 0 ⇒ y^{2} = 0 ⇒ y = 0, 0. Thus y-axis meets the parabola in two coincident points and hence y-axis touches the parabola at (0, 0).

(vi) As x → ∞ ⇒ y^{2} → ∞ ⇒ y → ±∞

Thus the curve is not bounded (closed) on the right side of the y-axis.

→ Double Ordinate: A chord passing through a point P on the parabola and perpendicular to the principal axis of the parabola is called the double ordinate of the point P.

→ Focal Chord: A chord of the parabola passing through the focus is called a focal chord.

→ Latus Rectum: A focal chord of a parabola perpendicular to the principal axis of the parabola is called latus rectum. If the latus rectum meets the parabola in L and L’, then LL’ is called length of the latus rectum.

Theorem: The length of the latus rectum of the parabola y^{2} = 4ax is 4a.

Proof:

Let LL’ be the length of the latus rectum of the parabola y^{2} = 4ax.

Let SL = 1, then L = (a, 1)

Since L is a point on the parabola y^{2} = 4ax, therefore 1^{2} = 4a(a)

⇒ 1^{2} = 4a^{2} ⇒ 1 = 2a ⇒ SL = 2a

∴ LL’ = 2SL = 4a.

→ Focal Distance: If P is a point on the parabola with focus S, then SP is called focal distance of P.

Theorem: The focal distance of P(x_{1}, y_{1}) on the parabola y^{2} = 4ax is x_{1} + a.

Notation: We use the following notation in this chapter

S ≡ y^{2} – 4ax

S_{1} ≡ YY_{1} – 2a(x + x_{1})

S_{11} = S(x_{1}, y_{1}) ≡ y_{1}^{2} – 4ax_{1}

S_{12} ≡ y_{1}y_{2} – 2a(x_{1} + x_{2})

Note:

Let P(x_{1}, y_{1}) be a point and S ≡ y^{2} – 4ax = 0 be a parabola. Then

- P lies on the parabola ⇔ S
_{11}= 0 - P lies inside the parabola ⇔ S
_{11}= 0 - P lies outside the parabola ⇔ S
_{11}= 0

Theorem: The equation of the chord joining the two points A(x_{1}, y_{1}), B(x_{2}, y_{2}) on the parabola S = 0 is S_{1} + S_{2} = S_{12}.

Theorem: The equation of the tangent to the parabola S = 0 at P(x_{1}, y_{1}) is S_{1} = 0.

Normal:

Let S = 0 be a parabola and P be a point on the parabola S = 0. The line passing through P and perpendicular to the tangent of S = 0 at P is called the normal to the parabola S = 0 at P.

Theorem: The equation of the normal to the parabola y^{2} = 4ax at P(x_{1}, y_{1}) is y_{1}(x – x_{1}) + 2a(y – y_{1}) = 0.

Proof:

The equation of the tangent to S = 0 at P is S_{1} = 0

⇒ yy_{1} – 2a(x + x_{1}) = 0.

⇒ yy_{1} – 2ax – 2ax_{1} = 0

The equation of the normal to S = 0 at P is:

y_{1}(x – x_{1}) + 2a(y – y_{1}) = 0

Theorem: The condition that the line y = mx + c may be a tangent to the parabola y^{2} = 4ax is c = a/m.

Proof:

Equation of the parabola is y^{2} = 4ax ……….. (1)

Equation of the line is y = mx + c ………. (2)

Solving (1) and (2),

(mx + c)^{2} = 4ax ⇒ m^{2}x^{2} + c^{2} + 2mcx = 4ax

⇒ m^{2}x^{2} + 2(mc – 2a)x + c^{2} = 0

Which is a quadratic equation in x. Therefore it has two roots.

If (2) is a tangent to the parabola, then the roots of the above equation are equal.

⇒ its disc eminent is zero

⇒ 4(mc – 2a)^{2} – 4m^{2}c^{2} = 0

⇒ m^{2}c^{2} + 4a^{2} – 4amc – m^{2}c^{2} = 0

⇒ a^{2} – amc = 0

⇒ a = mc

⇒ C = \(\frac{a}{m}\)

II Method:

Given parabola is y^{2} = 4ax.

Equation of the tangent is y = mx + c ———— (1)

Let P(x_{1}, y_{1}) be the point of contact.

The equation of the tangent at P is

yy_{1} – 2a(x + x_{1}) = 0 ⇒ yy_{1} = 2ax + 2ax_{1} ……. (2)

Now (1) and (2) represent the same line.

∴ \(\frac{\mathrm{y}_{1}}{1}=\frac{2 \mathrm{a}}{\mathrm{m}}=\frac{2 \mathrm{ax}_{1}}{\mathrm{c}}\) ⇒ x_{1} = \(\frac{\mathrm{c}}{\mathrm{m}}\), y_{1} = \(\frac{\mathrm{2a}}{\mathrm{m}}\)

P lies on the line y = mx + c ⇒ y_{1} = mx_{1} + c

⇒ \(\frac{2 \mathrm{a}}{\mathrm{m}}=\mathrm{m}\left(\frac{\mathrm{c}}{\mathrm{m}}\right)+\mathrm{c} \Rightarrow \frac{2 \mathrm{a}}{\mathrm{m}}=2 \mathrm{c} \Rightarrow \mathrm{c}=\frac{\mathrm{a}}{\mathrm{m}}\)

Note: The equation of a tangent to the parabola y^{2} = 4ax can be taken as y = mx + a/m. And the point of contact is (a/m^{2}, 2a/m).

Corollary: The condition that the line lx + my + n = 0 to touch the parabola y^{2} = 4ax is am^{2} = ln.

Proof:

Equation of the parabola is y^{2} = 4ax …………. (1)

Equation of the line is lx + my + n = 0

⇒ y = – \(\frac{1}{\mathrm{~m}}\)x – \(\frac{\mathrm{n}}{\mathrm{m}}\)

But this line is a tangent to the parabola, therefore

C = a/m ⇒ \(-\frac{\mathrm{n}}{\mathrm{m}}=\frac{\mathrm{a}}{-1 / \mathrm{m}} \Rightarrow \frac{\mathrm{n}}{\mathrm{m}}=\frac{\mathrm{am}}{1}\) ⇒ am^{2} = ln

Hence the condition that the line lx + my + n = 0 to touché the parabola y^{2} = 4ax is am^{2} = ln.

Note: The point of contact of lx + my + n = 0 with y^{2} = 4ax is (n/l, – 2am/l).

Theorem: The condition that the line lx + my + n = 0 to touch the parabola x^{2} = 4ax is al^{2} = mn.

Proof:

Given line is lx + my + n = 0 …… (1)

Let P(x_{1}, y_{1}) be the point of contact of (1) with the parabola x^{2} = 4ay.

The equation of the tangent at P to the parabola is xx_{1} = 2a(y + y_{1})

⇒ x_{1}x – 2ay – 2ay_{1} = 0 …… (2)

Now (1) and (2) represent the same line.

∴ \(\frac{\mathrm{x}_{1}}{1}=-\frac{2 \mathrm{a}}{\mathrm{m}}=-\frac{2 \mathrm{ay}_{1}}{\mathrm{n}}\) ⇒ x_{1} = – \(-\frac{2 \mathrm{al}}{\mathrm{m}}\), y_{1} = \(\frac{n}{m}\)

P lies on the line lx + my + n = 0

⇒ lx_{1} + my_{1} + n = 0 ⇒ l\(\left(\frac{-2 a l}{m}\right)\) + m\(\left(\frac{\mathrm{n}}{\mathrm{m}}\right)\) + n = 0

⇒ – 2al^{2} + mn + mn = 0 ⇒ al^{2} = mn

Theorem: Two tangents can be drawn to a parabola from an external point.

Note:

1. If m_{1}, m_{2} are the slopes of the tangents through P, then m_{1}, m_{2} become the roots of

equation (1). Hence m_{1} + m_{2} = y_{1}/x_{1}, m_{1}m_{2} = a/x_{1}.

2. If P is a point on the parabola S =0 then the roots of equation (1) coincide and hence only one tangent can be drawn to the parabola through P.

3. If P is an internal point to the parabola S = 0 then the roots of (1) are imaginary and hence no tangent can be drawn to the parabola through P.

Theorem: The equation in the chord of contact of P(x_{1}, y_{1}) with respect to the parabola S = 0 is S_{1} = 0.

Theorem: The equation of the chord of the parabola S = 0 having P(x_{1}, y_{1}) as its midpoint is S_{1} = S_{11}.

Pair of Tangents:

Theorem: The equation to the pair of tangents to the parabola S = 0 from P(x_{1}, y_{1}) is S_{1}^{2} = s_{11}s.

Parametric Equations of the Parabola:

A point (x, y) on the parabola y^{2} = 4ax can be represented as x = at^{2}, y = 2at in a single parameter t. Theses equations are called parametric equations of the parabola y^{2} = 4ax. The point (at^{2}, 2at) is simply denoted by t.

Theorem: The equation of the tangent at (at^{2}, 2at) to the parabola is y^{2} = 4ax is yt = x + at^{2}.

Proof:

Equation of the parabola is y^{2} = 4ax.

Equation of the tangent at (at^{2}, 2at) is S_{1} = 0.

⇒ (2at)y – 2a(x + at^{2}) = 0

⇒ 2aty = 2a(x + at^{2}) ⇒ yt = x + at^{2}.

Theorem: The equation of the normal to the parabola y^{2} = 4ax at the point t is y + xt = 2at + at^{3}.

Proof:

Equation of the parabola is y^{2} = 4ax.

The equation of the tangent at t is:

yt = x + at^{2} = x – yt + at^{2} = 0

The equation of the normal at (at^{2}, 2at) is

t(x – at^{2}) + l(y – 2at) = 0

⇒ xt – at^{3} + y – 2at = 0 ⇒ y + xt = 2at + at^{3}

Theorem: The equation of the chord joining the points t_{1} and t_{2} on the parabola y^{2} = 4ax is y(t_{1} + t_{2}) = 2x + 2at_{1}t_{2}.

Proof:

Equation of the parabola is y^{2} = 4ax.

Given points on the parabola are

P(at_{1}^{2}, 2at_{1}), Q(at_{2}^{2}, 2at_{2}) .

Slope of \(\overline{\mathrm{PQ}}\) is

\(\frac{2 \mathrm{at}_{2}-2 \mathrm{at}_{1}}{\mathrm{at}_{2}^{2}-\mathrm{at}_{1}^{2}}=\frac{2 \mathrm{a}\left(\mathrm{t}_{2}-\mathrm{t}_{1}\right)}{\mathrm{a}\left(\mathrm{t}_{2}^{2}-\mathrm{t}_{1}^{2}\right)}=\frac{2}{\mathrm{t}_{1}+\mathrm{t}_{2}}\)

The equation of \(\overline{\mathrm{PQ}}\) is y – 2at_{1} = \(\frac{2}{t_{1}+t_{2}}\) (x – at_{1}^{2}).

⇒ (y – 2at_{1}) (t_{1} + t_{2}) = 2(x – at_{1}^{2})

⇒ y(t_{1} + t_{2}) – 2at_{1}^{2} – 2at_{1}t_{2} = 2x – 2at_{1}^{2}

⇒ y(t_{1} + t_{2}) = 2x + 2at_{1}t_{2}.

Note: If the chord joining the points t_{1} and t_{2} on the parabola y^{2} = 4ax is a focal chord then t_{1}t_{2} = – 1.

Proof:

Equation of the parabola is y^{2} = 4ax

Focus S = (a, o)

The equation of the chord is y(t_{1} + t_{2}) = 2x + 2at_{1}t_{2}

If this is a focal chord then it passes through the focus (a, 0).

∴ 0 = 2a + 2at_{1}t_{2} ⇒ t_{1}t_{2} = – 1.

Theorem: The point of intersection of the tangents to the parabola y^{2} = 4ax at the

points t_{1} and t_{2} is (at_{1}t_{2}, a[t_{2} + t_{2}]).

Proof:

Equation of the parabola is y^{2} = 4ax

The equation of the tangent at t_{1} is yt_{1} = x + at_{1}^{2} ……. (1)

The equation of the tangent at t_{2} is

yt_{2} = x + at_{2}^{2} ……….. (2)

(1) – (2) ⇒ y(t_{1} – t_{2}) = a(t_{1}^{2} – t_{2}^{2}) ⇒ y = a(t_{1} + t_{2})

(1) ⇒ a(t_{1} + t_{2})t_{1} = x + at_{1}^{2}

= at_{1}^{2} + at_{1}t_{1} = x + at_{1}^{2} ⇒ x = at_{1}t_{2},

∴ Point of intersection = (at_{1}t_{2}, a[t_{1} + t_{2}]).

Theorem: Three normals can be drawn form a point (x_{1}, y_{1}) to the parabola y^{2} = 4ax.

Corollary: If the normal at t_{1} and t_{2}, to the parabola y^{2} = 4ax meet on the parabola, then t_{1}t_{2} = 2.

Proof:

Let the normals at t_{1} and t_{2} meet at t_{3} on the parabola.

The equation of the normal at t1 is:

y + xt_{1} = 2at_{1} + at_{1}^{3} ………… (1)

Equation of the chord joining t1 and t3 is:

y(t_{1} + t_{3}) = 2x + 2at_{1}t_{3} ……… (2)

(1) and (2) represent the same line

∴ \(\frac{t_{1}+t_{3}}{1}=\frac{-2}{t_{1}} \Rightarrow t_{3}=-t_{1}-\frac{2}{t_{1}}\)

Similarly t_{3} = – t_{1} – \(\frac{2}{\mathrm{t}_{2}}\)

∴ \(-\mathrm{t}_{1}-\frac{2}{\mathrm{t}_{1}}=-\mathrm{t}_{2}-\frac{2}{\mathrm{t}_{2}}\) ⇒ t_{1} – t_{2} \(\frac{2}{\mathrm{t}_{2}}-\frac{2}{\mathrm{t}_{1}}\)

⇒ t_{1} – t_{2} = \(\frac{2\left(\mathrm{t}_{1}-\mathrm{t}_{2}\right)}{\mathrm{t}_{1} \mathrm{t}_{2}}\) ⇒ t_{1}t_{2} = 2