JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let the tangent drawn to the parabola \(y ^{2}=24 x\) at the point \((\alpha, \beta)\) is perpendicular to the line \(2 x\) \(+2 y=5\). Then the normal to the hyperbola \(\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1\) at the point \((\alpha+4, \beta+4)\) does \(NOT\) pass through the point.
- A \((25,10)\)
- B \((20,12)\)
- C \((30,8)\)
- D \((15,13)\)
Answer & Solution
Correct Answer
(D) \((15,13)\)
Step-by-step Solution
Detailed explanation
Tangent at \((\alpha, \beta)\) has slope 1 \(\beta^{2}=24 \alpha\) Equation of tangent \(y \beta=12(x+\alpha), \frac{12}{\beta}=1\) \(\Rightarrow \alpha=6, \beta=12\) \(\therefore(\alpha+4, \beta+4)=(10,16)\) Normal at \((10,16)\) to \(\frac{x^{2}}{36}-\frac{y^{2}}{144}=1\) is…
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 \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by
\(f(x)=(2+3 a) x^2+\left(\frac{a+2}{a-1}\right) x+b, a \neq 1 . \text { If }\)
\(f(x+\mathrm{y})=f(x)+f(\mathrm{y})+1-\frac{2}{7} x \mathrm{y}\), then the value of \(28 \sum_{i=1}^5|f(i)|\) isJEE Mains 2025 Medium - The number of solutions of the equation \(\sin ^{-1}\left[x^{2}+\frac{1}{3}\right]+\cos ^{-1}\left[x^{2}-\frac{2}{3}\right]=x^{2}\) for \(x \in[-1,1],\) and \([x]\) denotes the greatest integer less than or equal to \(x\), is ...... .JEE Mains 2021 Hard
- Let for \(A=\left[\begin{array}{lll}1 & 2 & 3 \\ a & 3 & 1 \\ 1 & 1 & 2\end{array}\right],|A|=2\). If \(|2 \operatorname{adj}(2 \operatorname{adj}(2 A ))|\) \(=32^{ n }\), then \(3 n +\alpha\) is equal toJEE Mains 2023 Hard
- If the system of linear equations \(x + ky + 3z = 0;3x + ky - 2z = 0\) ; \(2x + 4y - 3z = 0\) has a non-zero solution \(\left( {x,y,z} \right)\) then \(\frac{{xz}}{{{y^2}}} = \). . . . .JEE Mains 2018 Hard
- If \(S=\{z \in C:|z-i|=|z+i|=|z-1|\}\), then, \(n(S)\) is:JEE Mains 2024 Medium
- Let \(f : R \rightarrow R\) be defined as \(f ( x )= x ^{3}+ x -5\). If \(g ( x )\) is a function such that \(f ( g ( x ))= x\), \(\forall x \in R\), then \(g ^{\prime}(63)\) is equal toJEE Mains 2022 Hard
More PYQs from JEE Mains
- If domain of the function \(\log _e\left(\frac{6 x^2+5 x+1}{2 x-1}\right)+\cos ^{-1}\left(\frac{2 x^2-3 x+4}{3 x-5}\right)\) is \((\alpha, \beta) \cup(\gamma, \delta]\), then \(18\left(\alpha^2+\beta^2+\gamma^2+\delta^2\right)\) is equal to \(....\).JEE Mains 2023 Hard
- If some three consecutive in the binomial expansion of \({\left( {x + 1} \right)^n}\) in powers of \(x\) are in the ratio \(2 : 15 : 70\), then the average of these three coefficient isJEE Mains 2019 Hard
- Let \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) be a thrice differentiable function such that \(f(0)=0, f(1)=1, f(2)=-1, f(3)=2\) and \(f(4)=-2\). Then, the minimum number of zeros of \(\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x)\) is ...........JEE Mains 2024 Hard
- The value of \( \int_{-\pi/6}^{\pi/6}(\frac{\pi+4x^{11}}{1-\sin(|x|+\pi/6)}) dx \) is equal to:JEE Mains 2026 Hard
- Let \(f(x)=\log _{\mathrm{e}} x\) and \(g(x)=\frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1}\). Then the domain of \(f \circ g\) isJEE Mains 2025 Medium
- If the line \(y\, = \,mx\, + \,7\sqrt 3 \) is normal to the hyperbola \(\frac{{{x^2}}}{{24}} - \frac{{{y^2}}}{{18}} = 1,\) then a value of \(m\) isJEE Mains 2019 Hard