JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let \(H _{ n }=\frac{ x ^2}{1+ n }-\frac{ y ^2}{3+ n }=1, n \in N\). Let \(k\) be the smallest even value of \(n\) such that the eccentricity of \(H _{ k }\) is a rational number. If \(l\) is length of the latus return of \(H _{ k }\), then \(21 l\) is equal to \(.......\).
- A \(305\)
- B \(306\)
- C \(304\)
- D \(303\)
Answer & Solution
Correct Answer
(B) \(306\)
Step-by-step Solution
Detailed explanation
\(Hn \Rightarrow \frac{ x ^2}{1+ n }-\frac{ y ^2}{3+ n }=1\) \(e =\sqrt{1+\frac{ b ^2}{ a ^2}}=\sqrt{1+\frac{3+ n }{1+ n }}=\sqrt{\frac{2 n +4}{ n +1}}\) \(e =\sqrt{\frac{2 n +4}{ n +1}}\) \(n =48(\text { smallest even value for which } e \in Q )\) \(e =\frac{10}{7}\)…
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
- If \(y(x)=\left|\begin{array}{ccc}\sin x & \cos x & \sin x+\cos x+1 \\ 27 & 28 & 27 \\ 1 & 1 & 1\end{array}\right|, x \in \mathbb{R}\), then \(\frac{d^2 y}{d x^2}+y\) is equal toJEE Mains 2025 Easy
- The sum of the coefficients of \(x^{499}\) and \(x^{500}\) in \((1+x)^{1000}+x(1+x)^{999}+x^{2}(1+x)^{998}+.......+x^{1000}\) isJEE Mains 2026 Easy
- If the equation of the normal to the curve \(y=\frac{x-a}{(x+b)(x-2)}\) at the point \((1,-3)\) is \(x-4 y=13\), then the value of \(a+b\) is equal to \(.......\).JEE Mains 2023 Hard
- If the perpendicular bisector of the line segment joining the points \(P (1,4)\) and \(Q ( k , 3)\) has \(y\)- intercept equal to \(-4,\) then a value of \(k\) isJEE Mains 2020 Medium
- If the system of linear equations, \(x+y+z =6\) ; \(x+2 y+3 z =10\) ; \(3 x+2 y+\lambda z =\mu\) has more two solutions, then \(\mu-\lambda^{2}\) is equal toJEE Mains 2020 Hard
- Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, \(\frac{x^{2}}{4}+\frac{y^{2}}{2}=1\) from any of its foci?JEE Mains 2020 Hard
More PYQs from JEE Mains
- For some \(a, b, c \in N\), let \(f(x)=a x-3\) and \(g ( x )= x ^{ b }+ c , x \in R\). If \((\text { fog })^{-1}( x )=\left(\frac{ x -7}{2}\right)^{1 / 3}\) then \((fog) (ac) + (gof) (b)\) is equal to \(..........\)JEE Mains 2023 Hard
- The circumcentre of a triangle lies at the origin and its centroid is the mid point of the line segment joining the points \((a^2 + 1 , a^2 + 1 )\) and \((2a, - 2a)\), \(a \ne 0\). Then for any \(a\) , the orthocentre of this triangle lies on the lineJEE Mains 2014 Hard
- A line in the \(3-\) dimensional space makes an angle \(\theta \left( {0 < \theta \le \frac{\pi }{2}} \right)\) with both the \(x\) and \(y\) axes. Then the set ofall values of \(\theta \) is the intervalJEE Mains 2014 Hard
- Let the area of the region enclosed by the curves \(y=3 x, 2 y=27-3 x\) and \(y=3 x-x \sqrt{x}\) be \(A\). Then \(10 \mathrm{~A}\) is equal toJEE Mains 2024 Hard
- If the area of the region bounded by \(16x^2 - 9y^2 = 144\) and \(8x - 3y = 24\) is A, then \(3(A + 6 \log_e(3))\) is equal to _______.JEE Mains 2026 Hard
- If \(y=y(x)\) is solution of the differential equation \(x\frac{{dy}}{{dx}} + 2y = {x^2}\) satisfying \(y(1)=1\) then \(y\left( {\frac{1}{2}} \right)\) is equal toJEE Mains 2019 Hard