Maths-

General

Easy

Question

Let A and B be two points on a parabola $y to the power of 2 end exponent equals x$ with vertex V such that VA is perpendicular to VB and θ is the angle between the chord VA and the axis of the parabola The value of $fraction numerator vertical line V A vertical line over denominator vertical line V B vertical line end fraction$ is

$t a n θ$
${t a n}^{3} θ$
${c o t}^{2} θ$
${c o t}^{3} θ$

The correct answer is: $c o t cubed theta$

we are given with two points A and B and asked to find the value of $fraction numerator vertical line V A vertical line over denominator vertical line V B vertical line end fraction$
$y^{2} = x$ , vertex is at the $o r i g i n$ and axis is $x - a x i s$ .
Let A has co-ordinates $(a, b)$

Now, according to question

$t a n θ = a b$

Squaring both sides we get,

$⟹ t a n^{2} θ = a ^{2} b ^{2}$

we know that $b^{2} = a$

Hence, we get

$⟹ t a n^{2} θ = a 1$

$⟹ a = c o t^{2} θ$

$⟹ b = c o t θ$

Similarly, for B

its co-ordinates are { $c o t (90 - θ), c o t^{2} (90 - θ)$ } as $∠ A O B = 9 0^{\circ}$

$⟹ a = t a n^{2} θ$

$⟹ b = t a n θ$

Now, applying distance formula we get

$( ∣ V B ∣ ) ^{2} ( ∣ V A ∣ ) ^{2} = t a n ^{4} θ + t a n ^{2} θ c o t ^{4} θ + c o t ^{2} θ$

$( ∣ V B ∣ ) ^{2} ( ∣ V A ∣ ) ^{2} = t a n ^{4} θ ( c o t ^{2} θ + 1 ) c o t ^{2} θ ( c o t ^{2} θ + 1 )$

$( ∣ V B ∣ ) ^{2} ( ∣ V A ∣ ) ^{2} = c o t6 θ$

$( ∣ V B ∣ ) ( ∣ V A ∣ ) = c o t³ θ$

Therefore the value of $fraction numerator vertical line V A vertical line over denominator vertical line V B vertical line end fraction$ is cot³θ

The equation of the parabola whose vertex and focus lie on the axis of $x$ at distances a and $a subscript 1 end subscript$ from the origin, respectively, is

maths-General

General

Maths-

Consider, $L subscript 1 end subscript colon 2 x plus 3 y plus p minus 3 equals 0 semicolon L subscript 2 end subscript colon 2 x plus 3 y plus p plus 3 equals 0 comma$ where p is a real number, and $C colon x to the power of 2 end exponent plus y to the power of 2 end exponent plus 6 x minus 10 y plus 30 equals 0.$
Statement‐l:: If line $L subscript 1 end subscript$ is a chord of circle $C$ , then line $L subscript 2 end subscript$ is not always adiameter ofcircle C.
Statement‐II :: If line $L subscript 1 end subscript$ is a diameter ofcircle $C$ , then line $L subscript 2 end subscript$ is not a chord ofcircle C.

Maths-General

General

Maths-

The equation of the circle passing through the points (1, 0) and (0,1) and having the smallest radius is:

Therefore the correct option is Choice 3

The equation of the circle passing through the points (1, 0) and (0,1) and having the smallest radius is:

Maths-General

Therefore the correct option is Choice 3

General

Maths-

If a circle passes through the point (a, b) and cuts the circle $x to the power of 2 end exponent plus y to the power of 2 end exponent equals p to the power of 2 end exponent$ orthogonally, then the equation of the locus of its centre is‐

Hence locus of the given condition is optionD

If a circle passes through the point (a, b) and cuts the circle $x to the power of 2 end exponent plus y to the power of 2 end exponent equals p to the power of 2 end exponent$ orthogonally, then the equation of the locus of its centre is‐

Maths-General

Hence locus of the given condition is optionD

General

Maths-

The square of the length of tangent from (3, -4) on the circle $x to the power of 2 end exponent plus y to the power of 2 end exponent minus 4 x minus 6 y plus 3 equals 0$

Hence option C is correct

The square of the length of tangent from (3, -4) on the circle $x to the power of 2 end exponent plus y to the power of 2 end exponent minus 4 x minus 6 y plus 3 equals 0$

Maths-General

Hence option C is correct

General

physics-

A hollow sphere of mass M and radius r is immersed in a tank of water (density r_w). The sphere would float if it were set free. The sphere is tied to the bottom of the tank by two wires which makes angle 45° with the horizontal as shown in the figure. The tension T₁ in the wire is :

physics-General

General

physics-

A small wooden ball of density r is immersed in water of density $sigma$ to depth h and then released. The height H above the surface of water up to which the ball will jump out of water is

physics-General

General

physics-

A dumbbell is placed in water of density r. It is observed that by attaching a mass m to the rod, the dumbbell floats with the rod horizontal on the surface of water and each sphere exactly half submerged as shown in the figure. The volume of the mass m is negligible. The value of length l is

physics-General

General

physics-

A slender homogeneous rod of length 2L floats partly immersed in water, being supported by a string fastened to one of its ends, as shown. The specific gravity of the rod is 0.75. The length of rod that extends out of water is :

physics-General

General

physics-

Two cubes of size 1.0 m sides, one of relative density 0.60 and another of relative density = 1.15 are connected by weightless wire and placed in a large tank of water. Under equilibrium the lighter cube will project above the water surface to a height of

physics-General

General

physics-

An open-ended U-tube of uniform cross-sectional area contains water (density 1.0 gram/centimeter³) standing initially 20 centimeters from the bottom in each arm. An immiscible liquid of density 4.0 grams/ centimeter³ is added to one arm until a layer 5 centimeters high forms, as shown in the figure above. What is the ratio h₂/h₁ of the heights of the liquid in the two arms?

physics-General

General

physics-

A light semi cylindrical gate of radius R is piovted at its mid point O, of the diameter as shown in the figure holding liquid of density r. The force F required to prevent the rotation of the gate is equal to

physics-General

General

physics-

In the figure shown, the heavy cylinder (radius R) resting on a smooth surface separates two liquids of densities 2r and 3r. The height ‘h’ for the equilibrium of cylinder must be

physics-General

General

physics-

A liquid of mass 1 kg is filled in a flask as shown in figure. The force exerted by the flask on the liquid is (g = 10 m/s²)[Neglect atmospheric pressure]:

physics-General

General

physics-

An open cubical tank was initially fully filled with water. When the tank was accelerated on a horizontal plane along one of its side it was found that one third of volume of water spilled out. The acceleration was

physics-General

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Let A and B be two points on a parabola with vertex V such that VA is perpendicular to VB and θ is the angle between the chord VA and the axis of the parabola The value of is

The correct answer is:

Related Questions to study

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The equation of the parabola whose vertex and focus lie on the axis of at distances a and from the origin, respectively, is

Consider, where p is a real number, and Statement‐l:: If line is a chord of circle , then line is not always adiameter ofcircle C.Statement‐II :: If line is a diameter ofcircle , then line is not a chord ofcircle C.

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The correct answer is: $c o t cubed theta$

The equation of the parabola whose vertex and focus lie on the axis of $x$ at distances a and $a subscript 1 end subscript$ from the origin, respectively, is

The equation of the parabola whose vertex and focus lie on the axis of $x$ at distances a and $a subscript 1 end subscript$ from the origin, respectively, is

If a circle passes through the point (a, b) and cuts the circle $x to the power of 2 end exponent plus y to the power of 2 end exponent equals p to the power of 2 end exponent$ orthogonally, then the equation of the locus of its centre is‐

If a circle passes through the point (a, b) and cuts the circle $x to the power of 2 end exponent plus y to the power of 2 end exponent equals p to the power of 2 end exponent$ orthogonally, then the equation of the locus of its centre is‐

The square of the length of tangent from (3, -4) on the circle $x to the power of 2 end exponent plus y to the power of 2 end exponent minus 4 x minus 6 y plus 3 equals 0$

The square of the length of tangent from (3, -4) on the circle $x to the power of 2 end exponent plus y to the power of 2 end exponent minus 4 x minus 6 y plus 3 equals 0$

A hollow sphere of mass M and radius r is immersed in a tank of water (density r_w). The sphere would float if it were set free. The sphere is tied to the bottom of the tank by two wires which makes angle 45° with the horizontal as shown in the figure. The tension T₁ in the wire is :

A hollow sphere of mass M and radius r is immersed in a tank of water (density r_w). The sphere would float if it were set free. The sphere is tied to the bottom of the tank by two wires which makes angle 45° with the horizontal as shown in the figure. The tension T₁ in the wire is :

A small wooden ball of density r is immersed in water of density $sigma$ to depth h and then released. The height H above the surface of water up to which the ball will jump out of water is

A small wooden ball of density r is immersed in water of density $sigma$ to depth h and then released. The height H above the surface of water up to which the ball will jump out of water is

A liquid of mass 1 kg is filled in a flask as shown in figure. The force exerted by the flask on the liquid is (g = 10 m/s²)[Neglect atmospheric pressure]:

A liquid of mass 1 kg is filled in a flask as shown in figure. The force exerted by the flask on the liquid is (g = 10 m/s²)[Neglect atmospheric pressure]: