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Maths-

General

Easy

Question

The differential equation of all ellipse centred at the origin and major and minor axes along coordinate axes is

$y^{2} + x y_{1}^{2} - y y_{1} = 0$
$x y y_{2} + x y_{1}^{2} - y y_{1} = 0$
$y y_{2} + x y_{1}^{2} - x y_{1} = 0$
None of these

The correct answer is: $x y y subscript 2 end subscript plus x y subscript 1 end subscript superscript 2 end superscript minus y y subscript 1 end subscript equals 0$

The given family of curves is
$fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ (i)
where a and b are parameters
$fraction numerator 2 x over denominator a to the power of 2 end exponent end fraction plus fraction numerator 2 y over denominator b to the power of 2 end exponent end fraction y subscript 1 end subscript equals 0 rightwards double arrow fraction numerator x over denominator a to the power of 2 end exponent end fraction plus fraction numerator y over denominator b to the power of 2 end exponent end fraction y subscript 1 end subscript equals 0$ (ii)
Differentiating again, we have
$fraction numerator 1 over denominator a to the power of 2 end exponent end fraction plus fraction numerator 1 over denominator b to the power of 2 end exponent end fraction open parentheses y subscript 1 end subscript superscript 2 end superscript plus y y subscript 2 end subscript close parentheses equals 0$ (iii)
Multiplying (iii) with x and then subtracting from (ii), we have
$open parentheses 1 divided by b to the power of 2 end exponent close parentheses open parentheses y y subscript 1 end subscript minus x y subscript 1 end subscript superscript 2 end superscript minus x y y subscript 2 end subscript close parentheses equals 0 rightwards double arrow x y y subscript 2 end subscript plus x y subscript 1 end subscript superscript 2 end superscript minus y y subscript 1 end subscript equals 0$

Related Questions to study

Area of the region bounded by $y equals tan invisible function application x comma$ tangent drawn to the curve at $x equals fraction numerator pi over denominator 4 end fraction$ and the $x minus$ axis is

Area of the region bounded by $y equals tan invisible function application x comma$ tangent drawn to the curve at $x equals fraction numerator pi over denominator 4 end fraction$ and the $x minus$ axis is

Maximum value of $x left parenthesis 1 minus x right parenthesis to the power of 2 end exponent$ , when $0 less or equal than x less or equal than 2$ , is

Maximum value of $x left parenthesis 1 minus x right parenthesis to the power of 2 end exponent$ , when $0 less or equal than x less or equal than 2$ , is

If x = $c o s to the power of negative 1 end exponent$ t, y = $square root of 1 minus t to the power of 2 end exponent end root$ , then $y subscript 2 end subscript equals$

If x = $c o s to the power of negative 1 end exponent$ t, y = $square root of 1 minus t to the power of 2 end exponent end root$ , then $y subscript 2 end subscript equals$

parallel

Two capacitors $C subscript 1 end subscript equals 2 mu F$ and $C subscript 2 end subscript equals 6 mu F$ in series, are connected in parallel to a third capacitor $C subscript 3 end subscript equals 4 mu F$ . This arrangement is then connected to a battery of e.m.f. = 2V, as shown in the figure. How much energy is lost by the battery in charging the capacitors

Two capacitors $C subscript 1 end subscript equals 2 mu F$ and $C subscript 2 end subscript equals 6 mu F$ in series, are connected in parallel to a third capacitor $C subscript 3 end subscript equals 4 mu F$ . This arrangement is then connected to a battery of e.m.f. = 2V, as shown in the figure. How much energy is lost by the battery in charging the capacitors

Physics-General

The combination of capacitors with $C subscript 1 end subscript equals 3 mu F comma C subscript 2 end subscript equals 4 mu F$ and $C subscript 3 end subscript equals 2 mu F$ is charged by connecting AB to a battery. Consider the following statements
I. Energy stored in $C subscript 1 end subscript$ = Energy stored in $C subscript 2 end subscript$ + Energy stored in $C subscript 3 end subscript$
II. Charge on C₁ = Charge on C₂ + Charge on C₃
III. Potential drop across C₁ = Potential drop across C₂ = Potential drop across C₃
Which of these is/are correct

The combination of capacitors with $C subscript 1 end subscript equals 3 mu F comma C subscript 2 end subscript equals 4 mu F$ and $C subscript 3 end subscript equals 2 mu F$ is charged by connecting AB to a battery. Consider the following statements
I. Energy stored in $C subscript 1 end subscript$ = Energy stored in $C subscript 2 end subscript$ + Energy stored in $C subscript 3 end subscript$
II. Charge on C₁ = Charge on C₂ + Charge on C₃
III. Potential drop across C₁ = Potential drop across C₂ = Potential drop across C₃
Which of these is/are correct

Physics-General

Consider a parallel plate capacitor of $10 mu rightwards arrow over short leftwards arrow F$ (micro-farad) with air filled in the gap between the plates. Now one half of the space between the plates is filled with a dielectric of dielectric constant 4, as shown in the figure. The capacity of the capacitor changes to

Consider a parallel plate capacitor of $10 mu rightwards arrow over short leftwards arrow F$ (micro-farad) with air filled in the gap between the plates. Now one half of the space between the plates is filled with a dielectric of dielectric constant 4, as shown in the figure. The capacity of the capacitor changes to

Physics-General

parallel

In the figure a capacitor is filled with dielectrics. The resultant capacitance is

In the figure a capacitor is filled with dielectrics. The resultant capacitance is

Physics-General

Equivalent capacitance between A and B is

Equivalent capacitance between A and B is

Physics-General

In the figure, three capacitors each of capacitance 6 $p F$ are connected in series. The total capacitance of the combination will be

In the figure, three capacitors each of capacitance 6 $p F$ are connected in series. The total capacitance of the combination will be

Physics-General

parallel

Two capacitors A and B are connected in series with a battery as shown in the figure. When the switch S is closed and the two capacitors get charged fully, then

Two capacitors A and B are connected in series with a battery as shown in the figure. When the switch S is closed and the two capacitors get charged fully, then

Physics-General

A parallel plate capacitor of area A, plate separation d and capacitance C is filled with three different dielectric materials having dielectric constants $k subscript 1 end subscript comma k subscript 2 end subscript$ and $k subscript 3 end subscript$ as shown. If a single dielectric material is to be used to have the same capacitance C in this capacitor, then its dielectric constant k is given by

A parallel plate capacitor of area A, plate separation d and capacitance C is filled with three different dielectric materials having dielectric constants $k subscript 1 end subscript comma k subscript 2 end subscript$ and $k subscript 3 end subscript$ as shown. If a single dielectric material is to be used to have the same capacitance C in this capacitor, then its dielectric constant k is given by

Physics-General

In the circuit here, the steady state voltage across capacitor C is a fraction of the battery e.m.f. The fraction is decided by

In the circuit here, the steady state voltage across capacitor C is a fraction of the battery e.m.f. The fraction is decided by

Physics-General

parallel

What is the effective capacitance between A and B in the following figure

What is the effective capacitance between A and B in the following figure

Physics-General

In the circuit shown in figure, each capacitor has a capacity of $3 mu F$ . The equivalent capacity between A and B is

In the circuit shown in figure, each capacitor has a capacity of $3 mu F$ . The equivalent capacity between A and B is

Physics-General

The capacitance between the points A and B in the given circuit will be

The capacitance between the points A and B in the given circuit will be

Physics-General

parallel

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