Students must practice this TS Inter 1st Year Maths 1B Study Material Chapter 10 Applications of Derivatives Ex 10(c) to find a better approach to solving the problems.

## TS Inter 1st Year Maths 1B Applications of Derivatives Solutions Exercise 10(c)

I.

Question 1.

Find the length of subtangent and sub¬normal at a point of the curve y = b sin\(\frac{x}{a}\).

Answer:

Equation of the curve is y = b sin\(\left(\frac{x}{a}\right)\)

\(\frac{d y}{d x}\) = b cos\(\left(\frac{x}{a}\right) \cdot \frac{1}{a}=\frac{b}{a}\) cos\(\left(\frac{x}{a}\right)\)

Question 2.

Show that the length of the subnormal at any point in the curve xy = a^{2} varies as the cube of the ordinate of the point. (V.S.A.Q.)

Answer:

Equation of the curve is xy = a^{2}

∴ Length of the subnormal αy_{1}^{3} = cube of the ordinate.

Question 3.

Show that at any point (x, y) on the curve y = be^{x/a}, the length of the subtangent is a constant and the length of the subnormal is \(\frac{y^2}{a}\). (V.S.A.Q)

Answer:

Equation of the given curve is y = be^{x/a}

∴ \(\frac{d y}{d x}=\frac{b}{a}\)e^{x/a} = \(\frac{\mathrm{y}}{\mathrm{a}}\)

∴ Slope at any point P(x, y) = \(\frac{y}{a}\)

Length of the subtangent = |y_{1}/f'(x_{1})|

= |y/\(\frac{y}{a}\)| = a = constant

Length of the subnormal = |y_{1}/f'(x_{1})|

= |y.\(\frac{y}{a}\)| = \(\left|\frac{y^2}{a}\right|\)

II.

Question 1.

Find the value of k so that the length of the subnormal at any point on the curve xy^{k} = a^{k+1} is a constant. (S.A.Q.)

Answer:

Equation of the curve is xy^{k} = a^{k+1}

Let P(xj, yO be any point on the curve then

x_{1}y_{1}^{k} = a^{k+1} …………….(1)

Differentiating w.r. to x,

Length of the subnormal is constant at any point on the curve is independent of x_{1} and y_{1}.

\(\frac{\mathrm{y}_1^{\mathrm{k}+2}}{\mathrm{k} \cdot \mathrm{a}^{\mathrm{k}+1}}\) is independent of x_{1} y_{1}.

∴ k + 2 = 0

⇒ k = – 2

Question 2.

At any point t on the curve x = a (t + sin t), y = a (1 – cos t), find the lengths of tangent, normal, subtangent and subnormal. (S.A.Q.) (June 2004, Board Model Paper)

Answer:

Equation of the curve is

x = a (t + sin t), y = a (1 – cos t)

Question 3.

Find the length of normal and subnormal at a point on the curve y = \(\frac{a}{2}\left(e^{\frac{x}{a}}+e^{\frac{-x}{a}}\right)\) (S.AQ.) (March 2013)

Answer:

Equation of the curve is y = \(\frac{a}{2}\left(e^{\frac{x}{a}}+e^{\frac{-x}{a}}\right)\)

= a cos h\(\left(\frac{x}{a}\right)\)

Question 4.

Find the lengths of subtangent, subnormal at a point’t’ on the curve x = a (cos t + t sin t), y = a (sin t – t cos t) (May 2014) (S.A.Q.)

Answer:

Equation of the curve is

x = a (cos t + t sin t)

y = a (sin t – t cos t)

\(\frac{\mathrm{dx}}{\mathrm{dt}}\) = a(-sint + tcost + sint) = at cos t

and \(\frac{\mathrm{dy}}{\mathrm{dt}}\)= a (cost + tsint – cost) = at sin t dt

∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dy}}{\mathrm{dt}} / \frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{at} \sin \mathrm{t}}{\mathrm{at} \cos \mathrm{t}}\) = tan t

= \(\left|\frac{\mathrm{a}(\sin \mathrm{t}-\mathrm{t} \cos \mathrm{t})}{\tan \mathrm{t}}\right|\)

= |a cot t(sin t – t cos t)|

Length of the subnormal = |y_{1}. f'(x_{1})|

= |a(sin t – t cos t) tan t|

= |a tan t(sin t – t cos t)|