JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
If the tangent to the parabola \(y^2 = x\) at a point \(\left( {\alpha ,\beta } \right)\,,\,\left( {\beta > 0} \right)\) is also a tangent to the ellipse, \(x^2 + 2y^2 = 1\), then \(a\) is equal to
- A \(2\sqrt 2 + 1\)
- B \(\sqrt 2 - 1\)
- C \(\sqrt 2 + 1\)
- D \(2\sqrt 2 - 1\)
Answer & Solution
Correct Answer
(C) \(\sqrt 2 + 1\)
Step-by-step Solution
Detailed explanation
Equation of tangent to the parabola \({y^2} = x\) \(At\left( {\alpha ,\beta } \right)\) is \(T=0\) \(y\beta = \frac{{x + \alpha }}{2}\) \( \Rightarrow y\beta = \frac{{x + {\beta ^2}}}{2}\,\) (\(\because\) \({\beta ^2} = \alpha \))…
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 \(\alpha=\sum_{k=0}^n\left(\frac{\left({ }^n C_k\right)^2}{k+1}\right)\) and \(\beta=\sum_{k=0}^{n-1}\left(\frac{{ }^n C_k{ }^n C_{k+1}}{k+2}\right)\). If \(5 \alpha=6 \beta\), then \(n\) equalsJEE Mains 2024 Hard
- Let \(f:[0, \infty) \rightarrow \mathbb{R}\) be differentiable function such that \(f(\mathrm{x})=1-2 \mathrm{x}+\int_0^x e^{x-t} f(t) \mathrm{dt}\) for all \(\mathrm{x} \in[0, \infty)\).
Then the area of the region bounded by \(\mathrm{y}=f(\mathrm{x})\) and the coordinate axes isJEE Mains 2025 Medium - If \(S\) is the sum of the first \(10\) terms of the series \(\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{13}\right)+\tan ^{-1}\left(\frac{1}{21}\right)+\ldots\) then \(\tan ( S )\) is equal toJEE Mains 2020 Medium
- If the in centre of an equilateral triangle is \((1, 1)\) and the equation of its one side is \(3x + 4y + 3\,= 0\), then the equation of the circumcircle of this triangle isJEE Mains 2015 Hard
- Let \(f(x)=2 \cos ^{-1} x+4 \cot ^{-1} x-3 x^{2}-2 x+10, x \in[-\) \(1,1]\). If \([ a , b ]\) is the range of the function then \(4 a -\) \(b\) is equal toJEE Mains 2022 Medium
- If \(\tan 15^{\circ}+\frac{1}{\tan 75^{\circ}}+\frac{1}{\tan 105^{\circ}}+\tan 195^{\circ}=2 a\), then the value of \(\left(a+\frac{1}{a}\right)\) is :JEE Mains 2023 Hard
More PYQs from JEE Mains
- Let \(y = y ( x )\) be the solution of the differential equation \(\left( x ^2-3 y ^2\right) dx +3 xy dy =0, y (1)=1\). Then \(6 y^2(e)\) is equal toJEE Mains 2023 Medium
- The sum of the square of the modulus of the elements in the set \(\{z=a+i b: a, b \in Z, z \in C,|z-1| \leq 1,|z-5| \leq|z-5 i|\}\) is ........JEE Mains 2024 Hard
- If \(A\) is a \(3×3\) non-singular matrix such that \(AA’=A’A \) and \( B=A^{-1}A’\) then \(BB’ \) equalsJEE Mains 2014 Medium
- The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, isJEE Mains 2025 Medium
- 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 :JEE Mains 2024 Hard
- If \(z=\frac{1}{2}-2 i\), is such that \(|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}\) and \(\alpha, \beta \in R \quad\), then \(\alpha+\beta\) is equal toJEE Mains 2024 Hard