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F* theta^ * is the angle between the circles passing through (lambda, 2lambda), (2lambda, lambda) and touching x-axis, the locus of the point from which pair of tangents drawn to the circle x ^ 2 y ^ 2 - 2x - 2y - 7 = 0 which includes the ang|e^ * theta * is x^ 2 y^ 2 -2x-2y-k= 0 number of positive integral divisors of 'k'?
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F* theta^ * is the angle between the circles passing through (lambda, ...
Problem:
F*θ* is the angle between the circles passing through (λ, 2λ), (2λ, λ) and touching x-axis, the locus of the point from which pair of tangents drawn to the circle x^2y^2 - 2x - 2y - 7 = 0 which includes the angle *θ* is x^2y^2 - 2x - 2y - k = 0. Find the number of positive integral divisors of 'k'.
Solution:
Part 1: Finding F*θ*
- Let's find the equation of the circles passing through (λ, 2λ) and (2λ, λ) and touching the x-axis.
- The midpoint of the line joining (λ, 2λ) and (2λ, λ) is ((λ+2λ)/2, (2λ+λ)/2) = (3λ/2, 3λ/2). This point lies on the circle.
- Also, the radius of the circle is the distance between the midpoint and any of the given points. Thus, the radius is √2λ^2.
- Hence, the equation of the circle is (x-3λ/2)^2 + (y-3λ/2)^2 = 2λ^2.
- Let's find the equation of the tangent to this circle at (a, b).
- The slope of the tangent is -[(a-3λ/2)/(b-3λ/2)].
- Hence, the equation of the tangent is (a-3λ/2)(x-a) + (b-3λ/2)(y-b) = λ^2.
- Since the tangent touches the x-axis, we can substitute y=0 in the above equation and solve for x.
- We get x = (λ^2 - b^2 + 3λb)/(2a - 3λ).
- Similarly, we can find the other point of intersection of the tangent with the circle, (x', 0), where x' = (λ^2 - b^2 - 3λb)/(2a - 3λ).
- Let's find the distance between the points (a, b) and (x', 0).
- The distance is √[(a-x')^2 + b^2].
- Using the formula for x and x', we can simplify the distance as √[(4λ^2 - 4ab)/(2a-3λ)^2].
- Now, we can use the cosine rule to find cos(F*θ*) = [(2λ)^2 + (2λ)^2 - {(4λ^2 - 4ab)/(2a-3λ)^2}]/[2*2λ*2λ].
- Simplifying the above equation, we get cos(F*θ*) = [(2a-5λ)^2 + 4b^2 - 8aλ]/[8λ(a-λ)].
- Hence, F*θ* = cos^-1[(2a-5λ)^2 + 4b^2 - 8aλ]/[8λ(a-λ)].
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F* theta^ * is the angle between the circles passing through (lambda, 2lambda), (2lambda, lambda) and touching x-axis, the locus of the point from which pair of tangents drawn to the circle x ^ 2 y ^ 2 - 2x - 2y - 7 = 0 which includes the ang|e^ * theta * is x^ 2 y^ 2 -2x-2y-k= 0 number of positive integral divisors of 'k'?
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F* theta^ * is the angle between the circles passing through (lambda, 2lambda), (2lambda, lambda) and touching x-axis, the locus of the point from which pair of tangents drawn to the circle x ^ 2 y ^ 2 - 2x - 2y - 7 = 0 which includes the ang|e^ * theta * is x^ 2 y^ 2 -2x-2y-k= 0 number of positive integral divisors of 'k'? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about F* theta^ * is the angle between the circles passing through (lambda, 2lambda), (2lambda, lambda) and touching x-axis, the locus of the point from which pair of tangents drawn to the circle x ^ 2 y ^ 2 - 2x - 2y - 7 = 0 which includes the ang|e^ * theta * is x^ 2 y^ 2 -2x-2y-k= 0 number of positive integral divisors of 'k'? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for F* theta^ * is the angle between the circles passing through (lambda, 2lambda), (2lambda, lambda) and touching x-axis, the locus of the point from which pair of tangents drawn to the circle x ^ 2 y ^ 2 - 2x - 2y - 7 = 0 which includes the ang|e^ * theta * is x^ 2 y^ 2 -2x-2y-k= 0 number of positive integral divisors of 'k'?.
F* theta^ * is the angle between the circles passing through (lambda, 2lambda), (2lambda, lambda) and touching x-axis, the locus of the point from which pair of tangents drawn to the circle x ^ 2 y ^ 2 - 2x - 2y - 7 = 0 which includes the ang|e^ * theta * is x^ 2 y^ 2 -2x-2y-k= 0 number of positive integral divisors of 'k'? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about F* theta^ * is the angle between the circles passing through (lambda, 2lambda), (2lambda, lambda) and touching x-axis, the locus of the point from which pair of tangents drawn to the circle x ^ 2 y ^ 2 - 2x - 2y - 7 = 0 which includes the ang|e^ * theta * is x^ 2 y^ 2 -2x-2y-k= 0 number of positive integral divisors of 'k'? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for F* theta^ * is the angle between the circles passing through (lambda, 2lambda), (2lambda, lambda) and touching x-axis, the locus of the point from which pair of tangents drawn to the circle x ^ 2 y ^ 2 - 2x - 2y - 7 = 0 which includes the ang|e^ * theta * is x^ 2 y^ 2 -2x-2y-k= 0 number of positive integral divisors of 'k'?.
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