Length of the subtangent at $(x_{l}, y_{l})$ on $x^{n} y^{m} = a^{m + n}, m, n > 0,$ is

\frac{n}{m} x_{l}

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\frac{m}{n} | x_{l} |

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\frac{n}{m} | y_{l} |

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\frac{n}{m} | x_{l} |

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Solution

The correct option is D $\frac{m}{n} | x_{l} |$
The subtangent is given by $\frac{y}{\frac{d y}{d x}}$
Here, $x^{n} y^{m} = a^{m + n}$
$∴ y^{m} = a^{m + n} x^{- n}$
$∴ y = (a^{\frac{m + n}{m}}) (x^{\frac{- n}{m}})$
$\Rightarrow \frac{d y}{d x} = (a^{\frac{m + n}{m}}) \times (\frac{- n}{m}) \times (x^{\frac{- n - m}{m}})$
$\Rightarrow$ subtangent at $(x_{l}, y_{l}) = a^{\frac{m + n}{m}} x_{l}^{\frac{- n}{m}} \times \frac{1}{(a^{\frac{m + n}{m}}) \times (\frac{- n}{m}) \times (x_{l}^{\frac{- n - m}{m}})}$ $= \frac{- m}{n} \times x_{l}$ $= \frac{m}{n} | x_{l} |$ , since length cannot be negative

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Differential coefficient of ${(x^{\frac{l + m}{m - n}})}^{\frac{1}{n - l}} . {(x^{\frac{m + n}{n - l}})}^{\frac{1}{l - m}} . {(x^{\frac{n + l}{l - m}})}^{\frac{1}{m - n}}$ w.r.t. $x$ is:

{(\frac{x^{m}}{x^{n}})}^{m + n} \times {(\frac{x^{n}}{x^{l}})}^{n + l} \times {(\frac{x^{l}}{x^{m}})}^{l + m}

Q. If

f (x) = {(\frac{x^{l}}{x^{m}})}^{l + m} {(\frac{x^{m}}{x^{n}})}^{m + n} {(\frac{x^{n}}{x^{l}})}^{n + 1}

, the f' (x) is equal to
(a) 1
(b) 0
(c)

x^{l + m + n}

(d) none of these

Q. (i) If

a = x^{m + n} y^{l}, b = x^{n + l} y^{m} and c = x^{l + m} y^{n}

, prove that

a^{m - n} b^{n - l} c^{l - m} = 1

.

(ii) If

x = a^{m + n}, y = a^{n + l} and z = a^{l + m}

, prove that

x^{m} y^{n} z^{l} = x^{n} y^{l} z^{m}

\int \frac{x^{3} + 1}{\sqrt{x^{2} + x}} d x

\frac{1}{k} [m \sqrt{x^{l} + x} + n l o g ∣ ∣ ∣ (x + \frac{1}{p}) + \sqrt{x^{r} + x} ∣ ∣ ∣] + C

.
Find

k + m + l + n + p + r

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Length of the subtangent at (xl,yl) on xnym=am+n,m,n>0,is

The correct option is D mn|xl|The subtangent is given by ydydxHere, xnym=am+n∴ym=am+nx−n∴y=(am+nm)(x−nm)⇒dydx=(am+nm)×(−nm)×(x−n−mm)⇒ subtangent at (xl,yl)=am+nmx−nml×1(am+nm)×(−nm)×(x−n−mml)=−mn×xl=mn|xl| , since length cannot be negative

Length of the subtangent at $(x_{l}, y_{l})$ on $x^{n} y^{m} = a^{m + n}, m, n > 0,$ is