JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let O be the origin, and P and Q be two points on the rectangular hyperbola \(xy = 12\) such that the mid point of the line segment PQ is \(\left(\dfrac{1}{2}, -\dfrac{1}{2}\right)\). Then the area of the triangle OPQ equals:
- A \(\dfrac{3}{2}\)
- B \(\dfrac{5}{2}\)
- C \(\dfrac{7}{2}\)
- D \(\dfrac{9}{2}\)
Answer & Solution
Correct Answer
(C) \(\dfrac{7}{2}\)
Step-by-step Solution
Detailed explanation
The equation of the chord of the hyperbola \(xy = 12\) with midpoint \((x_1, y_1)\) is given by \(T = S_1\). \(\dfrac{x y_1 + y x_1}{2} - 12 = x_1 y_1 - 12\) \(x y_1 + y x_1 = 2 x_1 y_1\) Substituting the midpoint \(\left(\dfrac{1}{2}, -\dfrac{1}{2}\right)\):…
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
- Consider the system of linear equations \(-x+y+2 z=0\) \(3 x-a y+5 z=1\) \(2 x-2 y-a z=7\) Let \(S_{1}\) be the set of all \(\mathrm{a} \in {R}\) for which the system is inconsistent and \(S_{2}\) be the set of all \(a \in {R}\) for which the system has infinitely many solutions. If \(n\left(S_{1}\right)\) and \(n\left(S_{2}\right)\) denote the number of elements in \(S_{1}\) and \(\mathrm{S}_{2}\) respectively, thenJEE Mains 2021 Hard
- If \(I(x)=\int e^{\sin ^2 x}(\cos x \sin 2 x-\sin x) d x \quad\) and \(I(0)=1\), then \(I\left(\frac{\pi}{3}\right)\) is equal toJEE Mains 2023 Hard
- The product of the roots of the equation \(9 x^{2}-18|x|+5=0,\) isJEE Mains 2020 Medium
- The least value of n for which the number of integral terms in the Binomial expansion of \((\sqrt[3]{7}+\sqrt[12]{11})^{\mathrm{n}}\) is 183, is :JEE Mains 2025 Medium
- The number of solutions of \(\tan^{-1}4x+\tan^{-1}6x=\frac{\pi}{6}\) where \(-\frac{1}{2\sqrt{6}}< x <\frac{1}{2\sqrt{6}}\) is equal toJEE Mains 2026 Easy
- 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
More PYQs from JEE Mains
- If the system of equations:
\(x+y+z=5\)
\(x+2y+3z=9\)
\(x+3y+\lambda z=\mu\)
has infinitely many solutions, then the value of \(\lambda+\mu\) is:JEE Mains 2026 Medium - Let \(f: R \rightarrow R\) be a function defined \(f(x)=\frac{2 e^{2 x}}{e^{2 x}+\varepsilon}\). Then \(f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+\ldots .+f\left(\frac{99}{100}\right)\) is equal toJEE Mains 2022 Hard
- A chord is drawn through the focus of the parabola \(y^2\, = 6x\) such that its distance from the vertex of this parabola is \(\frac{{\sqrt 5 }}{2}\) , then its slope can be:JEE Mains 2014 Hard
- A biased die is marked with numbers \(2,4,8,16,32,32\) on its faces and the probability of getting a face with mark \(n\) is \(\frac{1}{n}\). If the die is thrown thrice, then the probability, that the sum of the numbers obtained is \(48\) , isJEE Mains 2022 Medium
- Let \(f(x)=\left|\begin{array}{ccc}a & -1 & 0 \\ a x & a & -1 \\ a x^{2} & a x & a\end{array}\right|, a \in R\). Then the sum of which the squares of all the values of a for \(2 f^{\prime}(10)-f^{\prime}(5)+100=0\) isJEE Mains 2022 Hard
- If \(\alpha x+\beta y=109\) is the equation of the chord of the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\), whose mid point is \(\left(\frac{5}{2}, \frac{1}{2}\right)\), then \(\alpha+\beta\) is equal to :JEE Mains 2025 Medium