Have a query? Ask a Tutor!!

Access to best educators and exclusive paper discussions

Offsite Campus Turito Championship

Can’t find what you’re looking for?

Our tutors are from top schools around the world, and they are waiting for you 24/7, anytime, anywhere.

Homework- One to One Solutions
Understand concepts to solve better
Get your doubts solved instantly

Maths-

General

Easy

Question

If the area of the triangle, included between the axes and any tangent to the curve $x y to the power of n end exponent equals a to the power of n plus 1 end exponent$ is constant, then the value of $n$ is

$- 1$
$- 2$
1
2

The correct answer is: 1

We have,
$x y to the power of n end exponent equals a to the power of n plus 1 end exponent rightwards double arrow y to the power of n end exponent plus n blank x y to the power of n minus 1 end exponent fraction numerator d y over denominator d x end fraction equals 0 rightwards double arrow fraction numerator d y over denominator d x end fraction equals negative fraction numerator y over denominator n x end fraction$

Let $open parentheses x subscript 1 end subscript comma blank y subscript 1 end subscript close parentheses$ be a point on $x blank y to the power of n end exponent equals a to the power of n plus 1 end exponent$ . The equation of the tangent at $left parenthesis x subscript 1 end subscript comma blank y subscript 1 end subscript right parenthesis$ is

$y equals y subscript 1 end subscript equals negative fraction numerator y subscript 1 end subscript over denominator n x subscript 1 end subscript end fraction open parentheses x minus x subscript 1 end subscript close parentheses$

This meets with the coordinate axes at $A open parentheses open parentheses n plus 1 close parentheses x subscript 1 end subscript comma blank 0 close parentheses$ and $B open parentheses 0 comma fraction numerator open parentheses n plus 1 close parentheses y subscript 1 end subscript over denominator n end fraction close parentheses$

$therefore$ Area of $increment O A B equals fraction numerator 1 over denominator 2 end fraction open parentheses n plus 1 close parentheses x subscript 1 end subscript bullet open parentheses fraction numerator n plus 1 over denominator n end fraction close parentheses y subscript 1 end subscript$

$rightwards double arrow$ Area of $increment O A B equals fraction numerator 1 over denominator 2 end fraction fraction numerator open parentheses n plus 1 close parentheses to the power of 2 end exponent over denominator n end fraction x subscript 1 end subscript y subscript 1 end subscript$ …(i)

Since $left parenthesis x subscript 1 end subscript comma blank y subscript 1 end subscript right parenthesis$ lies on $x y to the power of n end exponent equals a to the power of n plus 1 end exponent$

$therefore x subscript 1 end subscript y subscript 1 end subscript superscript n end superscript equals a to the power of n plus 1 end exponent open parentheses x subscript 1 end subscript comma blank y subscript 1 end subscript close parentheses rightwards double arrow x subscript 1 end subscript equals fraction numerator a to the power of n plus 1 end exponent over denominator y subscript 1 end subscript superscript n end superscript end fraction$

Putting the value of $x subscript 1 end subscript$ in (i), we get

Area of $increment O A B equals fraction numerator 1 over denominator 2 end fraction fraction numerator open parentheses n plus 1 close parentheses to the power of 2 end exponent over denominator n end fraction a to the power of n plus 1 end exponent y subscript 1 end subscript superscript negative n plus 1 end superscript$

This will be a constant, if $n equals 1$

Related Questions to study

The fixed point $P$ on the curve $y equals x to the power of 2 end exponent minus 4 x plus 5$ such that the tangent at $P$ is perpendicular to the line $x plus 2 y minus 7 equals 0$ is given by

The fixed point $P$ on the curve $y equals x to the power of 2 end exponent minus 4 x plus 5$ such that the tangent at $P$ is perpendicular to the line $x plus 2 y minus 7 equals 0$ is given by

The function $f open parentheses x close parentheses equals x to the power of 3 end exponent plus a x to the power of 2 end exponent plus b x plus c comma a to the power of 2 end exponent less or equal than 3 b blank h a s$

The function $f open parentheses x close parentheses equals x to the power of 3 end exponent plus a x to the power of 2 end exponent plus b x plus c comma a to the power of 2 end exponent less or equal than 3 b blank h a s$

How many real solutions does the equation $x to the power of 7 end exponent plus 14 x to the power of 5 end exponent plus 16 x to the power of 3 end exponent plus 30 x minus 560 equals 0$ have?

How many real solutions does the equation $x to the power of 7 end exponent plus 14 x to the power of 5 end exponent plus 16 x to the power of 3 end exponent plus 30 x minus 560 equals 0$ have?

parallel

If $y equals x to the power of n end exponent comma$ then the ratio of relative errors in $y$ and $x$ is

If $y equals x to the power of n end exponent comma$ then the ratio of relative errors in $y$ and $x$ is

If a and b are positive numbers such that $a greater than b comma$ then the minimum value of $a sec invisible function application blank theta minus b tan invisible function application theta open parentheses 0 less than theta less than fraction numerator pi over denominator 2 end fraction close parentheses$ is

If a and b are positive numbers such that $a greater than b comma$ then the minimum value of $a sec invisible function application blank theta minus b tan invisible function application theta open parentheses 0 less than theta less than fraction numerator pi over denominator 2 end fraction close parentheses$ is

The side of an equilateral triangle is $´ a ´$ units and is increasing at the rate of $lambda blank u n i t s divided by s e c$ . The rate of increase of its area is

The side of an equilateral triangle is $´ a ´$ units and is increasing at the rate of $lambda blank u n i t s divided by s e c$ . The rate of increase of its area is

parallel

In how many ways four men and three women may sit around a round table if all the women are together.

In how many ways four men and three women may sit around a round table if all the women are together.

In how many ways 5 persons can sit at a round table, if two of the persons does not sit together.

In how many ways 5 persons can sit at a round table, if two of the persons does not sit together.

The function $f open parentheses x close parentheses equals fraction numerator x over denominator 2 end fraction plus fraction numerator 2 over denominator x end fraction$ has local minimum at

The function $f open parentheses x close parentheses equals fraction numerator x over denominator 2 end fraction plus fraction numerator 2 over denominator x end fraction$ has local minimum at

parallel

On which of the following intervals is the function $f open parentheses x close parentheses equals 2 blank x to the power of 2 end exponent minus log invisible function application open vertical bar x close vertical bar comma x not equal to 0$ increasing?

On which of the following intervals is the function $f open parentheses x close parentheses equals 2 blank x to the power of 2 end exponent minus log invisible function application open vertical bar x close vertical bar comma x not equal to 0$ increasing?

A conducting circular loop is placed in a uniform magnetic field $0.04 blank T$ with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at $2 blank m m divided by s$ . The induced $e m f$ in the loop when the radius is $2 blank c m$ is

A conducting circular loop is placed in a uniform magnetic field $0.04 blank T$ with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at $2 blank m m divided by s$ . The induced $e m f$ in the loop when the radius is $2 blank c m$ is

Physics-General

198.The transformation ratio in the step-up transformer is

198.The transformation ratio in the step-up transformer is

Physics-General

parallel

Two coils $A$ and $B$ having turns 300 and 600 respectively are placed near each other, on passing a current of $3.0 blank a m p e r e$ in $A$ , the flux linked with $A$ is $1.2 cross times 10 to the power of negative 4 end exponent blank w e b e r$ and with $B$ it is $9.0 cross times 10 to the power of negative 5 end exponent w e b e r$ . The mutual inductance of the system is

Two coils $A$ and $B$ having turns 300 and 600 respectively are placed near each other, on passing a current of $3.0 blank a m p e r e$ in $A$ , the flux linked with $A$ is $1.2 cross times 10 to the power of negative 4 end exponent blank w e b e r$ and with $B$ it is $9.0 cross times 10 to the power of negative 5 end exponent w e b e r$ . The mutual inductance of the system is

Physics-General

The magnetic flux across a loop of resistance $10 capital omega$ is given by $ϕ equals 5 t to the power of 2 end exponent minus 4 t plus 1 blank w e b e r$ . How much current is induced in the loop after 0.2 sec

The magnetic flux across a loop of resistance $10 capital omega$ is given by $ϕ equals 5 t to the power of 2 end exponent minus 4 t plus 1 blank w e b e r$ . How much current is induced in the loop after 0.2 sec

Physics-General

A conducting square loop of side L and resistance R moves in its plane with a uniform velocity v perpendicular to one of its sides. A magnetic induction B constant in time and space, pointing perpendicular and into the plane of the loop exists everywhere with part of the loop outside the field, as shown in figure. The induced emf is

A conducting square loop of side L and resistance R moves in its plane with a uniform velocity v perpendicular to one of its sides. A magnetic induction B constant in time and space, pointing perpendicular and into the plane of the loop exists everywhere with part of the loop outside the field, as shown in figure. The induced emf is
Physics-General

parallel

card img

With Turito Academy.

card img

With Turito Foundation.

card img

Get an Expert Advice From Turito.

Turito Academy

card img

With Turito Academy.

Test Prep

card img

With Turito Foundation.