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**B.** **Tangents at the Origin **

If a curve passing through the origin be given by a rational integral algebraic equation, the equation of the tangent (or tangents) at the origin is obtained by equating to zero the terms of the lowest degree in the equation.

e.g., if the equation of a curve be x^{2} - 4y^{2} + x^{4} + 3x^{3}y + 3x^{2} y^{2} + y^{4} = 0, the tangents at the origin are given by x^{2} - 4y^{2} = 0 or x + 2y and x - 2y = 0.

In the curve x^{2} + y^{2} + ax + by = 0, ax + by = 0, is the equation of the tangent at the origin; and in the curve (x^{2} + y^{2})^{2 }= a^{2} (x^{2} - y^{2}), x^{2} - y^{2} = 0 is the equation of a pair of tangents at the origin.

If the equation of a curve be x^{2} + y^{2} + x^{3} + 3^{ }x^{2 }y - y^{3} = 0, the tangents at the origin are given by x^{2} - y^{2} = 0 i.e. x + y = 0 and x - y = 0

**C. Angle of intersection**

Angle of intersection** **between two curves is defined as the angle between the two tangents drawn to the two curves at their point of intersection . If the angle between two curves is 90° then they are called **ORTHOGONAL** curves.

**Ex.10 Find the angle between curves y ^{2} = 4x and y = e^{-x/2}**

**Sol.**

Let the curves intersect at point (x_{1},y_{1})

**Note : **here that we have not actually found the intersection point but geometrically we can see that the curves intersect.

**Ex.11 Show that the curves y = 2 sin ^{2}x and y = cos 2x intersect at π/6. What is their angle of intersection ?**

**Sol.** Given curves are y = 2 sin^{2} x ...(1)

and y = cos 2x ...(2)

Solving (1) and (2), we get 2 sin^{2} x = cos 2x

⇒ 1 – cos 2x = cos 2x ⇒ cos2x = 1/2 ⇒ cos π/3 ⇒ 2x = ± π/3

x = ±π/6 are the points of intersection

From (1), dy/dx = 4 sin x cos x = 2 sin 2x = m_{1} (say)

From (2) dy/dx = -2 sin 2x = m_{2} (say)

If angle of intersection is θ, then tanθ

∴

∴

**Ex.12 Show that the angle between the tangents at any point P and the line joining P to the origin `O' is the same at all points of the curve ln (x ^{2} + y^{2}) = c tan^{-1} (y/x) where c is constant.**

**Sol. **Let the point P(x, y) on the curve ln (x^{2} + y^{2}) = c tan^{-1} (y/x) where c is constant.

Differentiating both sides w.r.t. x, we get

Slope of OP = y/x = m_{2} (say)

Let the angle between the tangents at P and OP be θ

which is independent of x and y..

**Ex.13 Show tht the curves **** intersect orthogonally.**

**Sol. **

Subtracting (2) from (1), we get

Hence given curves intersect orthogonally.

**Ex.14 Prove that the curves xy = 4 and x ^{2} + y^{2} = 8 touch each other.**

**Sol.**

Equation of the given curves are xy = 4 ....(i) and x^{2} + y^{2 }= 8 ....(ii)

Putting the value fo y from (i) i (ii), we get x^{2} + 16/x^{2} = 8 or or x^{4} + 16 = 8x^{2}

or x^{4 }– 8x^{2} + 16 = 0 or (x^{2} – 4)^{2} = 0 or x^{2 }– 4 = 0 or x^{2} = 4

from (i) ; when x = 2, y = 2 and when x = –2, y = -2

Hence points of intersection of the two curves are (2, 2) and (–2, –2).

Slope of the tangent to the curve (i) at point (2, 2) ⇒ m_{1} = -2/2 = -1 ...(from iii)

Slope of tangent to the curve (ii) at point (2, 2) ⇒ m_{2} = -2/2 = -1....( from iv)

Since m_{3} = m_{4}, hence the two curves touch each other at (-2, -2). Thus curves (i) and (ii) touch each other.

Slope of tangent to curve (i),

Slope of tangent to curve (ii),

Since m_{3} = m_{4}, hence the two curves touch each other at (–2, –2). Thus curves (i) and (ii) touch each other.

**Ex.15 The gradient of the common tangent to the two curves y = x ^{2} - 5x + 6 & y = x^{2} + x + 1 is**

**(A) - 1/3 **

**(B) - 2/3 **

**(C) - 1 **

**(D) - 3**

**Sol.** y = ax + b on solving with both curves and putting D = 0 gives

a^{2} + 10a + 4b + 1 = 0 and a^{2} - 2a + 4b - 3 = 0 ⇒ a = -^{ }1/3 & b = 5/9

⇒ 3x + 9y = 5 ; point of contact (7/3, -^{ }2/9) & (-^{ }2/3, 7/9)

**D. Length of Tangent **

**Ex.16 What should be the value of n in the equation of curve y = a ^{1 - n}. x^{n}, so that the sub-normal may be of constant length ?**

**Sol. **Given curve is y = a^{1 - n} . x^{n}

Taking logarithm of both sides, we get, *l*n y = (1 - n) *l*n a + n *l*n x

Differentiating both sides w.r.tx, we get ...(1)

Lengths of sub-normal = y dy/dx = y .ny/x ....{from 1}

Since lengths of sub-normal is to be constant, so x should not appear in its value i.e., 2n – 1 = 0. n =1/2

**Ex.17 If the relation between sub-normal SN and sub-tangent ST at any point S on the curve**

**by ^{2} = (x + a)^{3} is p(SN) = q(ST)^{2}; then p/q is**

**(A) 8b/27 **

**(B) b **

**(C) 1 **

**(D) none of these**

**Sol.**

Let a point by (x_{0}, y_{0}) lying on the curve by = (x_{0} + a)^{3} ....(i)

(from equation (i))

**Ex.18 For the curve y = **** show that sum of lengths of tangent & subtangent at any point is proportional to coordinates of point of tangency.**

**Sol.**

Let point of tangency be (x_{1}, y_{1}) ⇒

tangent + subtangent =

Hence proved.

**Ex.19 Show that the segment of the tangent to the curve y = **** contained between the y-axis and point of tangency has a constant length.**

**Sol.**

Equation of tangent at ‘ø’ y – a ln cot ø/2 + a cos ø =

⇒y sin ø – a sinø ln cot ø/2 + a sin ø cos ø - x cosø + a sin ø cos ø

⇒x cos ø+ y sin ø = a sin ø ln cot ø/2

Point on y-axis P ≡ (0, a ln cot ø/2) and point of tangency

Q≡ (a sinø a ln cotø/2 – a cos ø)

**E. Solving Equations **

**Ex.20 For what values of c does the equation ln x = cx^{2} have exactly one solution ?**

**Sol.**

Let's start by graphing y = In x and y = cx^{2} for various values of c. We know that for c * 0, y = cx^{2} is a parabola that opens upward if c > 0 and downward if c < 0. Figure 1 shows the parabolas y = cx^{2} for several positive values of c. Most of them don't intersect y = In x at all and one intersect twice. We have the feeling that there must be a value of c (somewhere between 0.1 and 0.3) for which the curves intersect exactly once, as in Figure 2.

To find that particular value of c, we let 'a' be the x-coordinate of the single point of intersection. In other words, In a = ca^{2}, so 'a' is the unique solution of the given equation. We see from Figure 2 that the curves just touch, so they have a common tangent line when x = a. That means the curves y = In x and y=cx^{2} have the same slope when x = a. Therefore 1/a = 2ca

Solving the equation In a = ca^{2} and 1/a = 2ca

For negative values of c we have the situation illustrated in Figure 3: All parabolas y = cx^{2} with negative values of c intersect y = In x exactly once. And let's not forget about c = 0: The curve y = 0 x^{2} = 0 just he x-axis, which intersects y = In x exactly once.

To summarize, the required values of c are c=1/(2e) and c<0

**Ex.21 The set of values of p for which the equation px ^{2 }= ln^{ }x possess a single root is**

**Sol.**

for p ≤ 0, there is obvious one solution ; for p > 0 one root

⇒ the curves touch each .

2 px_{1} = 1/x_{1} ⇒ x_{1}^{2} = 1/2p ;

Also px_{1}^{2} = ln x_{1} ⇒ p (1/2p) = ln x_{1} ⇒ x_{1} = e^{1/2}

⇒ 2 p = 1/e ⇒ p = 1/2e . Hence p ∈ (- ∝, 0] U {1/2e}

**F. Shortest distance **

Shortest distance between two non-intersecting curves always along the common normal (wherever defined)

**Ex.22 Find the shortest distance between the line y = x - 2 and the parabola y = x ^{2} + 3x + 2.**

**Sol. **Let P(x_{1}, y_{1}) be a point closest to the line y = x - 2 then = slope of line

⇒ 2x_{1} + 3 = 1 ⇒ x_{1} = - 1 ⇒ y_{1} = 0 Hence point (-1, 0) is the closest and its perpendicular distance from the

line y = x - 2 will give the shortest distance ⇒ p =

**Ex.23 Let P be a point on the curve C _{1}: y = **

**Sol.** Note that C_{1} is a semicircle and C_{2} is a rectangular hyperbola.

PQ will be minimum if the normal at P on the semicircle is also a normal at Q on xy = 9

Let the normal at P be y = mx ....(1) (m > 0) solving it with xy = 9

differentiating xy = 9

∴ normal at P and Q is y = x

solving P(1, 1) and Q(3, 3)

(PQ)^{2} = d^{2} = 4 + 4 = 8

**G. Rate Measurement **

**Ex.24 A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall ?**

**Sol. **We first draw a diagram and label it as in Figure 1. Let x feet be the distance from the bottom of the ladder to the wall and y feet the distance from the top of the ladder to the ground. Note that x and y are both function of t (time). We are given that dx/dt = 1 ft/s and we are asked to find dy/dt when x = 6 ft (see Figure 2). In this problem, the relationship between x and y is given by the Pythagorean Theorem : x^{2} + y^{2} = 100

Differentiating each side with respect to t using the Chain Rule, we have

and solving this equation for the desired rate, we obtain

When x = 6, the Pythagorean Theorem gives y = 8 and so, substituting these values and dx/dt = 1,

we have

The fact that dy/dt is negative means thatthe distance from the top of the ladder to the ground is decreasing at a rate of 3/4 ft/s In other words, the top of the ladder is sliding down the wall at a rate of 3/4 ft/s .

**Ex.25 A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m ^{3}/min, find the rate at which the water level is rising when the water is 3 m deep.**

**Sol. **

We first sketch the cone and label it as in Figure. Let V, r, and h be the volume of the water, the radius of the surface, and the height at time t, where t is measured in minutes.

We are given that dV/dt = 2m^{3}/min and we are asked to find dh/dt when h is 3 m. The quantities V and h are related by the equation V = 1/3pr^{2}hBut it is the very useful to express V as a function of h alone.

In order to eliminate r, we use the similar triangles in Figure to write and the expression for V becomes

Substituting h = 3 m and dV/dt = 2m^{3}/min, we have

The water level is rising at a rate of 8/(9p) ≈ 0.28 m/min.

**Ex.26 A man walks along a straight path at the speed of 4 ft/s. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight ?**

**Sol. **We draw Figure and let x be the distance from the man to the point on the path closest to the searchlight. We let θ be the distance from the man to the point on the path closest to the searchlight and the perpendicular to the path.

We are given that dx/dt = 4 ft/s and are asked to find dq/dt when x = 15. The equation that relates

Differentiating each side with respect to t, we get dx/dt = 20sec^{2} θ dθ/dt

when x = 15, the length of the beam is 25, so cosθ = 4/5

The searchlight is rotating at a rate of 0.128 rad/s.

**Ex.27 Two men A and B start with velocities v at the same time from the junction of two roads inclined at 45º to each other. If they travel by different roads, find the rate at which they are being separated.**

**Sol. **

Let L and M be the positions of men A and B at any time t,

Let OL = x and LM = y. Then OM = x

given, dx/dt = v; to find dy/dt from ΔLOM,

∴ they are being separated from each other at the rate

**Ex.28 A variable triangle ABC in the xy plane has its orthocentre at vertex 'B', a fixed vertex 'A' at the origin & the third vertex 'C' restricted to lie on the parabola y = ** ** The point B starts at the point (0, 1) at time t = 0 & moves upward along the y axis at a constant velocity of 2 cm/sec. How fast is the area of the triangle increasing when t = 7/2 sec ?**

**Sol. **

**Ex.29 Find the approximate value of (1.999) ^{6}.**

**Sol. **

Let f(x) = x^{6}. Now, f(x + δx) – f(x) = f'(x) . δx = 6x^{5} δx

We may write, 1.999 = 2 – 0.001

Taking x = 2 and δx = –0.001, we have f(1.999) – f(2) = 6(2)^{5} × – 0.001

⇒ f(1.999) = f(2) – 6 × 32 × 0.001 = 64 – 64 × 0.003 = 64 × 0.997 = 63.808 (approx).

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