JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
If the equation of the hyperbola with foci \((4,2)\) and \((8,2)\) is \(3 x^2-y^2-\alpha x+\beta y+\gamma=0\), then \(\alpha+\beta+\gamma\) is equal to _____.
- A 140
- B 141
- C 142
- D 143
Answer & Solution
Correct Answer
(B) 141
Step-by-step Solution
Detailed explanation
Equation of hyperbola is \(\frac{(x-6)^2}{a^2}-\frac{(y-2)^2}{4-a^2}=1\) \(\Rightarrow\left(4-\mathrm{a}^2\right)(\mathrm{x}-6)^2-\mathrm{a}^2(\mathrm{y}-2)^2=\mathrm{a}^2\left(4-\mathrm{a}^2\right)\) comparing with \(3 x^2-y^2-\alpha x+\beta y+\gamma=0\), we get…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(x=-1\) and \(x=2\) be the critical points of the function \(\mathrm{f}(\mathrm{x})=\mathrm{x}^3+\mathrm{ax}^2+\mathrm{b} \log _{\mathrm{c}}|\mathrm{x}|+1, \mathrm{x} \neq 0\). Let \(m\) and \(M\) respectively be the absolute minimum and the absolute maximum values of \(f\) in the interval \(\left[-2,-\frac{1}{2}\right]\). Then \(|\mathrm{M}+m|\) is equal to (Take \(\log _{\mathrm{c}} 2=0.7\) ):JEE Mains 2025 Medium
- Consider the lines \(\mathrm{L}_1: \mathrm{x}-1=\mathrm{y}-2=\mathrm{z}\) and \(\mathrm{L}_2: \mathrm{x}-2=\mathrm{y}=\mathrm{z}-1\). Let the feet of the perpendiculars from the point \(\mathrm{P}(5,1,-3)\) on the lines \(\mathrm{L}_1\) and \(\mathrm{L}_2\) be \(Q\) and \(R\) respectively. If the area of the triangle PQR is A , then \(4 \mathrm{~A}^2\) is equal to :JEE Mains 2025 Hard
- Let the product of \(\omega_1=(8+i) \sin \theta+(7+4 i) \cos \theta\) and \(\omega_2=(1+8 i) \sin \theta+(4+7 i) \cos \theta\) be \(\alpha+i \beta\), \(\mathrm{i}=\sqrt{-1}\). Let p and q be the maximum and the minimum values of \(\alpha+\beta\) respectively.JEE Mains 2025 Medium
- Let \(f, g: N -\{1\} \rightarrow N\) be functions defined by \(f(a)=\alpha\), where \(\alpha\) is the maximum of the powers of those primes \(p\) such that \(p^{\alpha}\) divides \(a\), and \(g(a)=a+1\), for all \(a \in N -\{1\}\). Then, the function \(f+ g\) is.JEE Mains 2022 Hard
- The inverse function of \(f(\mathrm{x})=\frac{8^{2 \mathrm{x}}-8^{-2 \mathrm{x}}}{8^{2 \mathrm{x}}+8^{-2 \mathrm{x}}}, \mathrm{x} \in(-1,1),\) isJEE Mains 2020 Hard
- Let \({\left( {x + 10} \right)^{50}} + {\left( {x - 10} \right)^{50}} = {a_0} + {a_1}x + {a_2}{x^2} + .... + {a_{50}}{x^{50}}\) , for \(x \in R\); then \(\frac{{{a_2}}}{{{a_0}}}\) is equal toJEE Mains 2019 Hard
More PYQs from JEE Mains
- Let \(S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^2 x}+9^{\tan ^2 x}=10\right\}\) and \(\beta=\sum_{x \in S} \tan ^2\left(\frac{x}{3}\right)\), then \(\frac{1}{6}(\beta-14)^2\) is equal toJEE Mains 2023 Hard
- If \(a, b, c\) are sides of a scalene triangle, then the value of \(\left| \begin{array}{*{20}{c}}
a&b&c\\
b&c&a\\
c&a&b
\end{array} \right|\) isJEE Mains 2013 Hard - Consider the equation \(\mathrm{x}^2+4 \mathrm{x}-\mathrm{n}=0\), where \(\mathrm{n} \in[20,100]\) is a natural number. Then the number of all distinct values of \(n\), for which the given equation has integral roots, is equal toJEE Mains 2025 Easy
- Let the equation \(\mathrm{x}(\mathrm{x}+2)(12-\mathrm{k})=2\) have equal roots. Then the distance of the point \(\left(\mathrm{k}, \frac{\mathrm{k}}{2}\right)\) from the line \(3 x+4 y+5=0\) isJEE Mains 2025 Medium
- A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is:JEE Mains 2026 Hard
- The number of values of \(k\) for which the system of linear equations, \((k + 2) x + 10y = k,\,\,kx + (k + 3)y = k - 1\) has no solution, isJEE Mains 2018 Hard