Maths

Equation of tangent of P (2cosθ, sinθ) is
(cosθ)x + (2sinθ)y = 4
Solving equation of tangent with equation of tangents at major axis ends, i.e. x = -2 and x = 2
For point 'B' (at x=-2):
-2cosθ + 2sinθ y = 4 ⇒ y = (2+cosθ)/sinθ
B (-2, (2+cosθ)/sinθ)
For point 'C' (at x=2):
2cosθ + 2sinθ y = 4 ⇒ y = (2-cosθ)/sinθ
C (2, (2-cosθ)/sinθ)
Now BC is the diameter of circle
Equation of circle: (x+2) (x-2) + (y - (2+cosθ)/sinθ) (y - (2-cosθ)/sinθ) = 0
x²-4 + y² - (4/sinθ)y + (4-cos²θ)/sin²θ = 0
Check if (√3, 0) satisfies this:
(√3)²-4 + 0 - 0 + (4-cos²θ)/sin²θ = -1 + (3+sin²θ)/sin²θ = -1 + 3/sin²θ + 1 = 3/sin²θ > 0.
This doesn't seem to work for all θ. Let's re-examine the coordinates.
Equation of tangent is x/2 cosθ + y sinθ = 1.
B (-2, (1+cosθ)/sinθ) = (-2, cot (θ/2)
C (2, (1-cosθ)/sinθ) = (2, tan (θ/2)
Equation of circle: (x+2) (x-2) + (y-cot (θ/2) (y-tan (θ/2) = 0
x²-4 + y² - (cot (θ/2)+tan (θ/2)y + 1 = 0
x²+y² - (2/sinθ)y - 3 = 0.
(√3, 0) satisfies this: 3 - 3 = 0.

Some of the common mistakes that people usually make while using logarithmic differentiation have been mentioned below:

• Not Multiplying by y: After logarithmic differentiation, it is mandatory to multiply by y to solve for dy/dx?
• Incorrectly Applying the Chain Rule: Make sure that you have correctly used the chain rule whenever you are differentiating a logarithmic expression.
• Using Wrong Logarithm: It is always advisable to use the natural logarithm (ln) instead of logarithms with other bases.
• Ignoring Domain Restrictions: Natural logarithm is only defined for the positive real numbers; therefore, y>0 whenever you apply logarithmic differentiation.

Logarithmic differentiation is used in the following cases:

• Logarithmic differentiation is used with functions that have a variable in both base and exponents. In such a case, standard differentiation rules do not apply directly to such functions. This differentiation converts exponentiation into multiplication.
• Another area where logarithmic differentiation is used is with a function which is the product of a quotient of multiple terms. 
• Whenever a function has a complex combination of multiplication, division and exponentiation, logarithmic differentiation is preferred. This differentiation eases the complexity by converting multiplicative relations into additive ones that are easier to handle.