Let S be the set of all the natural numbers, for which the line <math> <mrow> <mfrac> <mrow> <mi>x</mi> </mrow> <mrow> <mi>a</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>y</mi> </mrow> <mrow> <mi>b</mi> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> </mrow> </math> is a tangent to the curve <math> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>x</mi> </mrow> <mrow> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>y</mi> </mrow> <mrow> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mrow> </math> at the point (a, b), ab <math> <mrow> <mo>≠</mo> <mn>0</mn> <mo>.</mo> </mrow> </math> &nbsp;Then:

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Maths Applications of Derivatives Maths Class 12th

Let S be the set of all the natural numbers, for which the line $\frac{x}{a} + \frac{y}{b} = 2$ is a tangent to the curve ${(\frac{x}{a})}^{n} + {(\frac{y}{b})}^{n} = 2$ at the point (a, b), ab $\neq 0 .$ Then:

Option 1 -

$S = ϕ$

Option 2 -

n(S) = 1

Option 3 -

$S = {2 k : k \in N}$

Option 4 -

S=N

2 Views | Posted 2 weeks ago

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1 Answer

Answered by

Raj Pandey | Contributor-Level 9

2 weeks ago

Correct Option - 4

Detailed Solution:

${(\frac{x}{a})}^{n} + {(\frac{y}{b})}^{n} = 2$

$\Rightarrow \frac{n}{a} {(\frac{x}{a})}^{n - 1} + \frac{n}{b} {(\frac{y}{b})}^{n - 1} \frac{d y}{d x} = 0$

$\Rightarrow \frac{d y}{d x} = - \frac{b}{a} {(\frac{b x}{a y})}^{n - 1}$

$\frac{d y}{d x_{(a, b)}} = - \frac{b}{a}$

So line always touches the given curve.

<math> <mrow> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mi>x</mi> </mrow> <mrow> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mi>y</mi> </mrow> <mrow> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mrow> </math> <math> <mrow> <mo>⇒</mo> <mfrac> <mrow> <mi>n</mi> </mrow> <mrow> <mi>a</mi> </mrow> </mfrac> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mi>x</mi> </mrow> <mrow> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mfrac> <mrow> <mi>n</mi> </mrow> <mrow> <mi>b</mi> </mrow> </mfrac> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mi>y</mi> </mrow> <mrow> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </mrow> </math>  <math> <mrow> <mo>⇒</mo> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mi>b</mi> </mrow> <mrow> <mi>a</mi> </mrow> </mfrac> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mi>b</mi> <mi>x</mi> </mrow> <mrow> <mi>a</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math> <math> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mrow> <mo> (</mo> <mrow> <mi>a</mi> <mo>, </mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msub> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mi>b</mi> </mrow> <mrow> <mi>a</mi> </mrow> </mfrac> </mrow> </math> So line always touches the given curve.

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Let S be the set of all the natural numbers, for which the line $\frac{x}{a} + \frac{y}{b} = 2$ is a tangent to the curve ${(\frac{x}{a})}^{n} + {(\frac{y}{b})}^{n} = 2$ at the point (a, b), ab $\neq 0 .$ Then:

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Let S be the set of all the natural numbers, for which the line x a + y b = 2 is a tangent to the curve ( x a ) n + ( y b ) n = 2 at the point (a, b), ab ≠ 0 . Then:

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Let S be the set of all the natural numbers, for which the line $\frac{x}{a} + \frac{y}{b} = 2$ is a tangent to the curve ${(\frac{x}{a})}^{n} + {(\frac{y}{b})}^{n} = 2$ at the point (a, b), ab $\neq 0 .$ Then: