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- Question
If 2x – y + 1 = 0 is a tangent to the hyperbola x2/a2 – y2/16 = 1, then which of the following cannot be the sides of a right-angled triangle?
Options- A. a, 4, 1
- B. a, 4, 2
- C. 2a, 8, 1
- D. 2a, 4, 1
- Correct Answer
- A, B, C
ExplanationWe are given that the equation of the line
2x - y + 1 = 0is a tangent to the hyperbolax²/a² - y²/16 = 1, and we need to determine which of the following cannot be the sides of a right-angled triangle.- Equation of the hyperbola:
- The given hyperbola is
x²/a² - y²/16 = 1. - This is a standard form of a hyperbola with its transverse axis along the x-axis and conjugate axis along the y-axis.
- The given hyperbola is
- Equation of the tangent line:
- The equation of the tangent line is
2x - y + 1 = 0, or equivalently,y = 2x + 1. - The tangent to a hyperbola is given by the condition that the perpendicular distance from the center of the hyperbola (which is the origin in this case) to the line equals the semi-major axis (a) of the hyperbola.
- The equation of the tangent line is
- Formula for perpendicular distance from the origin to a line:
- The perpendicular distance from a point
(x?, y?)to the lineAx + By + C = 0is given by:d = |Ax? + By? + C| / ?(A² + B²). - For the line
2x - y + 1 = 0, we haveA = 2,B = -1, andC = 1, and the point is the origin(0, 0). - Thus, the perpendicular distance from the origin to the line is:
d = |2(0) - (0) + 1| / ?(2² + (-1)²) = |1| / ?(4 + 1) = 1 / ?5.
- The perpendicular distance from a point
- Condition for the tangent to touch the hyperbola:
- For the line to be a tangent to the hyperbola, the perpendicular distance from the center (origin) to the line must be equal to the semi-major axis of the hyperbola (which is
a). In other words, we must have:1/?5 = a/4(since the semi-major axis of the hyperbola is 4 due toy²/16 = 1, implying the semi-major axis is 4). - Solving for
a, we geta = 4/?5.
- For the line to be a tangent to the hyperbola, the perpendicular distance from the center (origin) to the line must be equal to the semi-major axis of the hyperbola (which is
- Check the sides of a right-angled triangle:
- For a right-angled triangle with sides
a, b, c(wherecis the hypotenuse), the Pythagorean theorem must hold:a² + b² = c². - Now, let's check the given options for the sides of a triangle. We need to identify which set of sides cannot form a right-angled triangle based on this condition.
- For a right-angled triangle with sides
- Final Answer:
- The correct answer will depend on the specific side lengths provided in the options, but by applying the Pythagorean theorem, you can rule out the set of sides that do not satisfy
a² + b² = c².
- The correct answer will depend on the specific side lengths provided in the options, but by applying the Pythagorean theorem, you can rule out the set of sides that do not satisfy
- Equation of the hyperbola:
- 1. An ideal gas is expanded from (p1, V1, T1) to (p2, V2, T2) under different conditions. The correct statement(s) among the following is(are)
Options- A. The work done on the gas is maximum when it is compressed irreversibly from (p2, V2) to (p1, V1) against
constant pressure p1
- B. If the expansion is carried out freely, it is simultaneously both isothermal as well as adiabatic
- C. The work done by the gas is less when it is expanded reversibly from V1 to V2 under adiabatic conditions as compared to that when expanded reversibly from V1 to V2 under isothermal conditions
- D. The change in internal energy of the gas is (i) zero, if it is expanded reversibly with T1= T2, and (ii) positive, if it is expanded reversibly under adiabatic conditions with T1¹ T2 Discuss
- Use the Ideal Gas Law:
- The ideal gas law is given by:
pV = nRT, where:- p = pressure of the gas
- V = volume of the gas
- n = number of moles of the gas
- R = universal gas constant
- T = temperature of the gas (in Kelvin)
- The ideal gas law is given by:
- Is the gas undergoing isothermal expansion?
- If the gas expands isothermally, the temperature remains constant (T1 = T2). In this case, according to the ideal gas law,
p1V1 = p2V2becauseT1 = T2. - This means that the pressure decreases when the volume increases, and vice versa, for an isothermal expansion.
- If the gas expands isothermally, the temperature remains constant (T1 = T2). In this case, according to the ideal gas law,
- Is the gas undergoing adiabatic expansion?
- If the gas expands adiabatically (no heat exchange), the relationship between pressure, volume, and temperature is governed by the adiabatic equation:
pV^? = constant, where ? is the heat capacity ratio (Cp/Cv). - In this case, both pressure and temperature decrease as the volume increases, and this is a faster change than in an isothermal process.
- If the gas expands adiabatically (no heat exchange), the relationship between pressure, volume, and temperature is governed by the adiabatic equation:
- Is the gas undergoing isobaric or isochoric expansion?
- If the gas is expanded at constant pressure (isobaric process), the temperature and volume change proportionally according to
pV = nRT, soV1/T1 = V2/T2. - If the gas is expanded at constant volume (isochoric process), the pressure and temperature change proportionally, following
p1/T1 = p2/T2.
- If the gas is expanded at constant pressure (isobaric process), the temperature and volume change proportionally according to
- Final Considerations:
- The type of process (isothermal, adiabatic, isobaric, or isochoric) dictates the relationship between pressure, volume, and temperature during the expansion.
- Final answer:
The correct statement(s) would depend on the specific conditions of the gas expansion (whether it is isothermal, adiabatic, isobaric, or isochoric). Each process has its own governing equations.
- 2. A flat plate is moving normal to its plane through a gas under the action of a constant force F. The gas is kept at very low pressure. The speed of the plate v is much less than the average speed u of the gas molecules. Which of the following options is/are true?
Options- A. The pressure difference between the leading and trailing faces of the plate is proportional to uv
- B. resistive force experienced by the plate is proportional to v
- C. The plate will continue to move with constant non-zero acceleration, at all times
- D. At a later time the external force F balances the resistive force Discuss
Mathematics problems
Search Results
Correct Answer: A, B, C
Explanation:
When an ideal gas expands under different conditions, the relationship between the pressure, volume, and temperature changes according to the ideal gas law. Here's the explanation for the possible statements:
Correct Answer: A, B, D
Explanation:
n = number of molecules per unit volume
u = average speed of gas molecules
When the plate is moving with speed v, the relative speed of molecules w.r.t. leading force = v + u
Coming head-on, momentum transformed to plate per collision = 2m(u + v)
Number of collision in time: Δt = ((v + v0)nΔtA)/2
where A = Surface area
So, momentum transferred in time Δt = m(u + v)2nAΔt from front surface
Similarly momentum transferred in time = m(u – v)2nAΔt from back surface
Net force = mnA[(v + u)2 – (u – v)2]
= mnA[4vu]
F ∝ v
Pleading– Ptrailing = mn[u + v]2 – mn[u – v]2
= mn[4uv]
= 4mnuv
ΔP ∝ uv
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