Circle Through A(2,−1), B(3,4), Center on (x−5)²+(y−1)²=13/2, Find r²
Circle C passes through A(2,−1) and B(3,4), AB not a diameter, and its centre lies on (x−5)²+(y−1)²=13/2. Find r² using the perpendicular-bisector property.
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Circle x²+y²=8x Meets Hyperbola x²/9−y²/4=1 at A,B — Centroid of PAB
A and B are intersection points of a circle and hyperbola; P moves on 2x−3y+4=0. Find the line on which the centroid of triangle PAB always lies.
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Binomial (1+x)^(n+4): 5th, 6th, 7th Coefficients in AP, Find Largest Coefficient
For some n≠10, the 5th, 6th and 7th coefficients of (1+x)^(n+4) are in AP. Find n, then the largest coefficient of the expansion.
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AP: a1+(a5+a10+…+a2020)+a2024=2233, Find Sum to 2024 Terms
For an AP a1,…,a2024 with a1+(a5+a10+…+a2020)+a2024=2233, find a1+a2+…+a2024 by summing the selected terms as their own smaller AP.
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A∩B for |(z+1)/(z−1)|<1 and arg((z−1)/(z+1))=2π/3
Let A: |(z+1)/(z−1)|<1 and B: arg((z−1)/(z+1))=2π/3 in the complex plane. Find A∩B by converting both conditions into geometric loci.
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S Is 10 Distinct Primes, A = Products of Two or More Elements, Find n(P)
S is a set of 10 distinct primes and A is the set of products of two or more elements of S. Find n(P) where P={(x,y): x∈S, y∈A, x divides y}.
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Midpoint of a Chord of the Ellipse x²/9+y²/4=1 Is (√2, 4/3), Find α
If the midpoint of a chord of the ellipse x²/9+y²/4=1 is (√2, 4/3), and the chord’s length is 2√α/3, then α is: (1) 18 (2) 22 (3) 26 (4) 20.
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Integrate cos x/[(4+sin²x)(5−4cos²x)]
Step-by-step solution to the integral of cos(x)/[(4+sin²x)(5−4cos²x)] dx, using a sine substitution and partial fractions to reduce it to two arctangent terms.
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f(x)=2x/√(1+9x²) Composed 10 Times, Find √(3α+1)
Consider f:R→R defined by f(x)=2x/√(1+9x²). If (f∘f∘⋯∘f, 10 times)(x)=2^10x/√(1+9αx²), find the value of √(3α+1).
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Sum of 1 + 6 + 9(1²+2²+3²)/7 + 12(1²+2²+3²+4²)/9 + … Up to 15 Terms
The sum of the series 1+6+9(1²+2²+3²)/7+12(1²+2²+3²+4²)/9+15(1²+…+5²)/11+… up to 15 terms is: (1)7820 (2)7520 (3)7830 (4)7510.