JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let \(H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1\) be the hyperbola, whose eccentricity is \(\sqrt{3}\) and the length of the latus rectum is \(4 \sqrt{3}\). Suppose the point \((\alpha, 6), \alpha>0\) lies on \(H\). If \(\beta\) is the product of the focal distances of the point \((\alpha, 6)\), then \(\alpha^2+\beta\) is equal to :
- A \(170\)
- B \(171\)
- C \(169\)
- D \(172\)
Answer & Solution
Correct Answer
(B) \(171\)
Step-by-step Solution
Detailed explanation
\( \mathrm{H}: \frac{\mathrm{y}^2}{\mathrm{~b}^2}-\frac{\mathrm{x}^2}{\mathrm{a}^2}=1, \mathrm{e}=\sqrt{3} \) \( \mathrm{e}=\sqrt{1+\frac{\mathrm{a}^2}{\mathrm{~b}^2}}=\sqrt{3} \quad \Rightarrow \frac{\mathrm{a}^2}{\mathrm{~b}^2}=2 \) \( a^2=2 b^2 \)…
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 \( f(\alpha) \) denote the area of the region in the first quadrant bounded by \( x=0, x=1, y^{2}=x \) and \( y=|\alpha x-5|-|1-\alpha x|+\alpha x^{2}. \) Then \( (f(0)+f(1)) \) is equal toJEE Mains 2026 Hard
- Let the points \(\left(\frac{11}{2}, \alpha\right)\) lie on or inside the triangle with sides \(x+y=11, x+2 y=16\) and \(2 x+3 y=29\). Then the product of the smallest and the largest values of \(\alpha\) is equal to :JEE Mains 2025 Medium
- If the lines \(\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}\) and \(\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}\) intersect, then the magnitude of the minimum value of \(8 \alpha \beta\) is \(...............\).JEE Mains 2023 Hard
- Let \(a , b , c\) be three distinct positive real numbers such that \((2 a)^{\log _{\varepsilon} a}=(b c)^{\log _e b}\) and \(b^{\log _e 2}=a^{\log _e c}\). Then \(6 a+5 b c\) is equal to \(........\).JEE Mains 2023 Hard
- A plane \(P\) contains the line \(x+2 y+3 z+1=0=x-y-z-6\) and is perpendicular to the plane \(-2 x+y+z+8=0\). Then which of the following points lies on \(\mathrm{P}\) ?JEE Mains 2021 Medium
- For the system of linear equations \(2 x+4 y+2 a z=b\) \(x+2 y+3 z=4\) \(2 x-5 y+2 z=8\) which of the following is NOT correct?JEE Mains 2023 Hard
More PYQs from JEE Mains
- If the system of linear equations \(2 \mathrm{x}+2 \mathrm{ay}+\mathrm{az}=0\) ; \(2 x+3 b y+b z=0\) ; \(2 \mathrm{x}+4 \mathrm{cy}+\mathrm{cz}=0\) ; where \(a, b, c \in R\) are non-zero and distinct; has a non-zero solution, thenJEE Mains 2020 Hard
- A straight line \(L\) at a distance of \(4\) units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of \(60^o\) with the line \(x + y = 0\). Then an equation of the line \(L\) isJEE Mains 2019 Hard
- If \(\sum_{\mathrm{r}=0}^{10}\left(\frac{10^{\mathrm{r}+1}-1}{10^{\mathrm{r}}}\right) \cdot{ }^{11} \mathrm{C}_{\mathrm{r}+1}=\frac{\alpha^{11}-11^{11}}{10^{10}}\), then \(\alpha\) is equal to :JEE Mains 2025 Hard
- Let : \(\overrightarrow{ a }=\hat{ i }+2 \hat{ j }+3 \hat{ k }, \overrightarrow{ b }=\hat{ i }-\hat{ j }+2 \hat{ k }\) and \(\vec{c}=5 \hat{i}-3 \hat{j}+3 \hat{k}\) be there vectors. If \(\vec{r}\) is a vector such that, \(\overrightarrow{ r } \times \overrightarrow{ b }=\overrightarrow{ c } \times \overrightarrow{ b }\) and \(\overrightarrow{ r } \cdot \overrightarrow{ a }=0\). Then \(25|\overrightarrow{ r }|^2\) is equal toJEE Mains 2023 Hard
- Let \(A =\{1,2,3,4,5,6,7\}\) and \(B =\{3,6,7,9\}\). Then the number of elements in the set \(\{ C \subseteq A : C \cap B \neq \phi\}\) isJEE Mains 2022 Medium
- Let \(P ( S )\) denote the power set of \(S =\{1,2,3, \ldots, 10\}\). Define the relations \(R_1\) and \(R_2\) on \(P(S)\) as \(A R_1 B\) if \(\left( A \cap B ^{ c }\right) \cup\left( B \cap A ^{ c }\right)=\varnothing\) and \(AR _2 B\) if \(A \cup B ^{ c }=\) \(B \cup A ^{ c }, \forall A , B \in P ( S )\). Then :JEE Mains 2023 Hard