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The minimum area of triangle formed by tangent to the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ and coordinate axes‐

ab
$\frac{a^{2} + b^{2}}{2}$
$\frac{(a + b)^{2}}{2}$
$\frac{a^{2} + a b + b^{2}}{3}$

The correct answer is: ab

Given, that a triangle is formed by tangent to the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ and coordinate axes, we need to find the minimum area of the triangle formed.
$E q u a t i o n space o f space tan g e n t space a t left parenthesis left parenthesis a cos left parenthesis empty set right parenthesis comma b sin left parenthesis empty set right parenthesis right parenthesis space t o space t h e space e l l i p s e space i s space x over a cos left parenthesis empty set right parenthesis plus y over b sin left parenthesis empty set right parenthesis equals 1 C o o r d i n a t e s space o f space P space a n d space Q space a r e space open parentheses fraction numerator a over denominator cos left parenthesis empty set right parenthesis end fraction comma 0 close parentheses space a n d open parentheses 0 comma fraction numerator b over denominator sin left parenthesis empty set right parenthesis end fraction close parentheses space r e s p e c t i v e l y. N o w comma space a r e a space o f space t r i a n g l e space O P Q space equals 1 half open vertical bar fraction numerator a over denominator cos left parenthesis empty set end fraction cross times fraction numerator b over denominator sin left parenthesis empty set right parenthesis end fraction close vertical bar equals space fraction numerator a b over denominator open vertical bar sin left parenthesis 2 empty set right parenthesis close vertical bar end fraction T h e r e f o r e space comma M i n i m u m space a r e a equals space a b$

Therefore, the minimum area is ab.

An ellipse has OB as semi minor axis, $F$ and $F to the power of † end exponent$ its focii and the angle FBF’ is a right angle Then the eccentricity of the ellipse is‐

Therefore, the eccentricity of the ellipse is $space space fraction numerator 1 over denominator square root of 2 end fraction$

An ellipse has OB as semi minor axis, $F$ and $F to the power of † end exponent$ its focii and the angle FBF’ is a right angle Then the eccentricity of the ellipse is‐

Maths-General

Therefore, the eccentricity of the ellipse is $space space fraction numerator 1 over denominator square root of 2 end fraction$

General

Maths-

The number of values of $c$ such that the straight line y=4x+c touches the curve $left parenthesis x to the power of 2 end exponent divided by 4 right parenthesis plus y to the power of 2 end exponent equals 1$ is‐

Therefore, there are two values of c.

The number of values of $c$ such that the straight line y=4x+c touches the curve $left parenthesis x to the power of 2 end exponent divided by 4 right parenthesis plus y to the power of 2 end exponent equals 1$ is‐

Maths-General

Therefore, there are two values of c.

General

Maths-

Let P be any point on any directrix of an ellipse Then the chords of contact of point P with respect to the ellipse and its auxiliary circle intersect at

Maths-General

General

maths-

An ellipse having foci at (3, 3) and (-4,4) and passing through the origin has eccentricity equal to‐

maths-General

General

maths-

Normals $A O comma A A subscript 1 end subscript comma A A subscript 2 end subscript$ are drawn to parabola $y to the power of 2 end exponent equals 8 x$ from the pointA (h, 0) If triangle $O A subscript 1 end subscript A subscript 2 end subscript$ ( $O$ being the origin) is equilateral, then possible value of h’ is

maths-General

General

maths-

Given: A circle, $2 x to the power of 2 end exponent plus 2 y to the power of 2 end exponent equals 5$ and a parabola, $y to the power of 2 end exponent equals 4 square root of 5 x$
Statement‐I:: An equation of a common tangent to these curves is $y equals x plus square root of 5.$
Statement‐II:: If the line, $y equals m x plus fraction numerator square root of 5 over denominator m end fraction left parenthesis m not equal to 0 right parenthesis$ is their common tangent, then $m$ satisfies $m to the power of 4 end exponent minus 3 m to the power of 2 end exponent plus 2 equals 0.$

maths-General

General

maths-

In any equilateral $capital delta$ , three circles of radii one are touching to the sides given as in the figure then area of the $capital delta$ is

maths-General

General

maths-

If the sides a, b, c of a triangle are such that a : b : c : : 1 : $square root of 3$ : 2, then the A : B : C is -

maths-General

General

physics-

A block placed on a rough inclined plane of inclination ( $theta$ =30°) can just be pushed upwards by applying a force "F" as shown. If the angle of inclination of the inclined plane is increased to ( $theta$ = 60°), the same block can just be prevented from sliding down by application of a force of same magnitude. The coefficient of friction between the block and the inclined plane is

physics-General

General

physics-

The tube AC forms a quarter circle in a vertical plane. The ball B has an area of cross–section slightly smaller than that of the tube, and can move without friction through it. B is placed at A and displaced slightly. It will

physics-General

General

physics-

A small cube with mass M starts at rest at point 1 at a height 4R, where R is the radius of the circular part of the track. The cube slides down the frictionless track and around the loop. The force that the track exerts on the cube at point 2 is nearly _____ times the cube's weight Mg.

physics-General

General

physics-

A bob attached to a string is held horizontal and released. The tension and vertical distance from point of suspension can be represented by.

physics-General

General

physics-

A ball whose size is slightly smaller than width of the tube of radius 2.5 m is projected from bottommost point of a smooth tube fixed in a vertical plane with velocity of 10 m/s. If N₁ and N₂ are the normal reactions exerted by inner side and outer side of the tube on the ball

physics-General

General

physics-

A block of mass m moving with a velocity v₀ on a smooth horizontal surface strikes and compresses a spring of stiffness k till mass comes to rest as shown in the figure. This phenomenon is observed by two observers:
A: standing on the horizontal surface
B: standing on the block

According to observer B, the potential energy of the spring increases

physics-General

General

physics-

A block of mass m moving with a velocity v0 on a smooth horizontal surface strikes and compresses a spring of stiffness k till mass comes to rest as shown in the figure. This phenomenon is observed by two observers:
A: standing on the horizontal surface
B: standing on the block

To an observer B, when the block is compressing the spring

physics-General

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Can’t find what you’re looking for?

ab
$\frac{a^{2} + b^{2}}{2}$
$\frac{(a + b)^{2}}{2}$
$\frac{a^{2} + a b + b^{2}}{3}$

The correct answer is: ab

Related Questions to study

An ellipse has OB as semi minor axis, $F$ and $F to the power of † end exponent$ its focii and the angle FBF’ is a right angle Then the eccentricity of the ellipse is‐

An ellipse has OB as semi minor axis, $F$ and $F to the power of † end exponent$ its focii and the angle FBF’ is a right angle Then the eccentricity of the ellipse is‐

The number of values of $c$ such that the straight line y=4x+c touches the curve $left parenthesis x to the power of 2 end exponent divided by 4 right parenthesis plus y to the power of 2 end exponent equals 1$ is‐

The number of values of $c$ such that the straight line y=4x+c touches the curve $left parenthesis x to the power of 2 end exponent divided by 4 right parenthesis plus y to the power of 2 end exponent equals 1$ is‐

Let P be any point on any directrix of an ellipse Then the chords of contact of point P with respect to the ellipse and its auxiliary circle intersect at

Let P be any point on any directrix of an ellipse Then the chords of contact of point P with respect to the ellipse and its auxiliary circle intersect at

An ellipse having foci at (3, 3) and (-4,4) and passing through the origin has eccentricity equal to‐

An ellipse having foci at (3, 3) and (-4,4) and passing through the origin has eccentricity equal to‐

In any equilateral $capital delta$ , three circles of radii one are touching to the sides given as in the figure then area of the $capital delta$ is

In any equilateral $capital delta$ , three circles of radii one are touching to the sides given as in the figure then area of the $capital delta$ is

If the sides a, b, c of a triangle are such that a : b : c : : 1 : $square root of 3$ : 2, then the A : B : C is -

If the sides a, b, c of a triangle are such that a : b : c : : 1 : $square root of 3$ : 2, then the A : B : C is -

The tube AC forms a quarter circle in a vertical plane. The ball B has an area of cross–section slightly smaller than that of the tube, and can move without friction through it. B is placed at A and displaced slightly. It will

The tube AC forms a quarter circle in a vertical plane. The ball B has an area of cross–section slightly smaller than that of the tube, and can move without friction through it. B is placed at A and displaced slightly. It will

A small cube with mass M starts at rest at point 1 at a height 4R, where R is the radius of the circular part of the track. The cube slides down the frictionless track and around the loop. The force that the track exerts on the cube at point 2 is nearly _____ times the cube's weight Mg.

A small cube with mass M starts at rest at point 1 at a height 4R, where R is the radius of the circular part of the track. The cube slides down the frictionless track and around the loop. The force that the track exerts on the cube at point 2 is nearly _____ times the cube's weight Mg.

A bob attached to a string is held horizontal and released. The tension and vertical distance from point of suspension can be represented by.

A bob attached to a string is held horizontal and released. The tension and vertical distance from point of suspension can be represented by.

A ball whose size is slightly smaller than width of the tube of radius 2.5 m is projected from bottommost point of a smooth tube fixed in a vertical plane with velocity of 10 m/s. If N₁ and N₂ are the normal reactions exerted by inner side and outer side of the tube on the ball

A ball whose size is slightly smaller than width of the tube of radius 2.5 m is projected from bottommost point of a smooth tube fixed in a vertical plane with velocity of 10 m/s. If N₁ and N₂ are the normal reactions exerted by inner side and outer side of the tube on the ball

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The minimum area of triangle formed by tangent to the ellipse and coordinate axes‐

ab

The correct answer is: ab

Given, that a triangle is formed by tangent to the ellipse and coordinate axes, we need to find the minimum area of the triangle formed.

Related Questions to study

An ellipse has OB as semi minor axis, and its focii and the angle FBF’ is a right angle Then the eccentricity of the ellipse is‐

An ellipse has OB as semi minor axis, and its focii and the angle FBF’ is a right angle Then the eccentricity of the ellipse is‐

The number of values of such that the straight line y=4x+c touches the curve is‐

The number of values of such that the straight line y=4x+c touches the curve is‐

Let P be any point on any directrix of an ellipse Then the chords of contact of point P with respect to the ellipse and its auxiliary circle intersect at

Let P be any point on any directrix of an ellipse Then the chords of contact of point P with respect to the ellipse and its auxiliary circle intersect at

An ellipse having foci at (3, 3) and (-4,4) and passing through the origin has eccentricity equal to‐

An ellipse having foci at (3, 3) and (-4,4) and passing through the origin has eccentricity equal to‐

Normals are drawn to parabola from the pointA (h, 0) If triangle ( being the origin) is equilateral, then possible value of h’ is

Normals are drawn to parabola from the pointA (h, 0) If triangle ( being the origin) is equilateral, then possible value of h’ is

Given: A circle, and a parabola, Statement‐I:: An equation of a common tangent to these curves is Statement‐II:: If the line, is their common tangent, then satisfies

Given: A circle, and a parabola, Statement‐I:: An equation of a common tangent to these curves is Statement‐II:: If the line, is their common tangent, then satisfies

In any equilateral , three circles of radii one are touching to the sides given as in the figure then area of the is

In any equilateral , three circles of radii one are touching to the sides given as in the figure then area of the is

If the sides a, b, c of a triangle are such that a : b : c : : 1 : : 2, then the A : B : C is -

If the sides a, b, c of a triangle are such that a : b : c : : 1 : : 2, then the A : B : C is -

The tube AC forms a quarter circle in a vertical plane. The ball B has an area of cross–section slightly smaller than that of the tube, and can move without friction through it. B is placed at A and displaced slightly. It will

The tube AC forms a quarter circle in a vertical plane. The ball B has an area of cross–section slightly smaller than that of the tube, and can move without friction through it. B is placed at A and displaced slightly. It will

A small cube with mass M starts at rest at point 1 at a height 4R, where R is the radius of the circular part of the track. The cube slides down the frictionless track and around the loop. The force that the track exerts on the cube at point 2 is nearly _____ times the cube's weight Mg.

A small cube with mass M starts at rest at point 1 at a height 4R, where R is the radius of the circular part of the track. The cube slides down the frictionless track and around the loop. The force that the track exerts on the cube at point 2 is nearly _____ times the cube's weight Mg.

A bob attached to a string is held horizontal and released. The tension and vertical distance from point of suspension can be represented by.

A bob attached to a string is held horizontal and released. The tension and vertical distance from point of suspension can be represented by.

A ball whose size is slightly smaller than width of the tube of radius 2.5 m is projected from bottommost point of a smooth tube fixed in a vertical plane with velocity of 10 m/s. If N1 and N2 are the normal reactions exerted by inner side and outer side of the tube on the ball

A ball whose size is slightly smaller than width of the tube of radius 2.5 m is projected from bottommost point of a smooth tube fixed in a vertical plane with velocity of 10 m/s. If N1 and N2 are the normal reactions exerted by inner side and outer side of the tube on the ball

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ab
$\frac{a^{2} + b^{2}}{2}$
$\frac{(a + b)^{2}}{2}$
$\frac{a^{2} + a b + b^{2}}{3}$

An ellipse has OB as semi minor axis, $F$ and $F to the power of † end exponent$ its focii and the angle FBF’ is a right angle Then the eccentricity of the ellipse is‐

An ellipse has OB as semi minor axis, $F$ and $F to the power of † end exponent$ its focii and the angle FBF’ is a right angle Then the eccentricity of the ellipse is‐

The number of values of $c$ such that the straight line y=4x+c touches the curve $left parenthesis x to the power of 2 end exponent divided by 4 right parenthesis plus y to the power of 2 end exponent equals 1$ is‐

The number of values of $c$ such that the straight line y=4x+c touches the curve $left parenthesis x to the power of 2 end exponent divided by 4 right parenthesis plus y to the power of 2 end exponent equals 1$ is‐

In any equilateral $capital delta$ , three circles of radii one are touching to the sides given as in the figure then area of the $capital delta$ is

In any equilateral $capital delta$ , three circles of radii one are touching to the sides given as in the figure then area of the $capital delta$ is

If the sides a, b, c of a triangle are such that a : b : c : : 1 : $square root of 3$ : 2, then the A : B : C is -

If the sides a, b, c of a triangle are such that a : b : c : : 1 : $square root of 3$ : 2, then the A : B : C is -

A ball whose size is slightly smaller than width of the tube of radius 2.5 m is projected from bottommost point of a smooth tube fixed in a vertical plane with velocity of 10 m/s. If N₁ and N₂ are the normal reactions exerted by inner side and outer side of the tube on the ball

A ball whose size is slightly smaller than width of the tube of radius 2.5 m is projected from bottommost point of a smooth tube fixed in a vertical plane with velocity of 10 m/s. If N₁ and N₂ are the normal reactions exerted by inner side and outer side of the tube on the ball