JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let C be the circle \(\mathrm{x}^2+(\mathrm{y}-1)^2=2, \mathrm{E}_1\) and \(\mathrm{E}_2\) be two ellipses whose centres lie at the origin and major axes lie on x -axis and y -axis respectively. Let the straight line \(x+y=3\) touch the curves \(C\), \(E_1\) and \(E_2\) at \(P\left(x_1, y_1\right), Q\left(x_2, y_2\right)\) and \(R\left(x_3, y_3\right)\) respectively. Given that \(P\) is the mid-point of the line segment \(Q R\) and \(P Q=\frac{2 \sqrt{2}}{3}\), the value of \(9\left(x_1 y_1+x_2 y_2+x_3 y_3\right)\) is equal to ______ .
- A 42
- B 44
- C 40
- D 46
Answer & Solution
Correct Answer
(D) 46
Step-by-step Solution
Detailed explanation
Let \(\mathrm{E}_1: \frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1,(\mathrm{a} \gt \mathrm{b})\) \(\mathrm{E}_2: \frac{\mathrm{x}^2}{\mathrm{c}^2}+\frac{\mathrm{y}^2}{\mathrm{~d}^2}=1,(\mathrm{c} \lt \mathrm{d})\) \(C: x^2+(y-1)^2=2\) Equation of tangent…
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 \(S\) be the set of all \((\lambda, \mu)\) for which the vectors \(\lambda \hat{ i }-\hat{ j }+\hat{ k }, \hat{ i }+2 \hat{ j }+\mu \hat{ k }\) and \(3 \hat{ i }-4 \hat{ j }+5 \hat{ k }\), where \(\lambda-\mu=5\), are coplanar, then \(\sum_{(\lambda, \mu) \in S} 80\left(\lambda^2+\mu^2\right)\) is equal to :JEE Mains 2023 Hard
- Let \(P \left(\frac{2 \sqrt{3}}{\sqrt{7}}, \frac{6}{\sqrt{7}}\right), Q , R\) and \(S\) be four points on the ellipse \(9 x^2+4 y^2=36\). Let \(P Q\) and \(RS\) be mutually perpendicular and pass through the origin. If \(\frac{1}{( PQ )^2}+\frac{1}{( RS )^2}=\frac{ p }{ q }\), where \(p\) and \(q\) are coprime, then \(p+q\) is equal to \(.........\).JEE Mains 2023 Hard
- The area (in square units) of the region bounded by the curves \(y + 2x^2 = 0\) and \(y + 3x^2 = 1\) , is equal toJEE Mains 2015 Hard
- If the line of intersection of the planes \(a x+b y=3\) and \(ax + by + cz =0, a >0\) makes an angle \(30^{\circ}\) with the plane \(y - z +2=0\), then the direction cosines of the line are.JEE Mains 2022 Hard
- A tangent line \(\mathrm{L}\) is drawn at the point \((2,-4)\) on the parabola \(\mathrm{y}^{2}=8 \mathrm{x}\). If the line \(\mathrm{L}\) is also tangent to the circle \(x^{2}+y^{2}=a\), then \('a'\) is equal to .... .JEE Mains 2021 Hard
- Let \(\vec{a}\) and \(\vec{b}\) be two vectors. Let \(|\vec{a}|=1,|\vec{b}|=4\) and \(\vec{a} \cdot \vec{b}=2\). If \(\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}\), then the value of \(\overrightarrow{ b } \cdot \overrightarrow{ c }\) isJEE Mains 2023 Medium
More PYQs from JEE Mains
- If a tangent to the ellipse \(x^{2}+4 y^{2}=4\) meets the tangents at the extremities of its major axis at \(\mathrm{B}\) and \(\mathrm{C}\), then the circle with \(\mathrm{BC}\) as diameter passes through the point:JEE Mains 2021 Hard
- If the solution of the equation \(\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1, x \in\left(0, \frac{\pi}{2}\right), \quad\) is \(\sin ^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)\), where \(\alpha, \beta\) are integers, then \(\alpha+\beta\) is equal to:JEE Mains 2023 Hard
- Let \(A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]\) If \(A^3=4 A^2-A-21 I\), where I is the identity matrix of order \(3 \times 3\), then \(2 a+3 b\) is equal to :JEE Mains 2024 Hard
- Let \(f(x)=\int_0^x\left(t+\sin \left(1-e^t\right)\right) d t, x \in \mathbb{R}\). Then \(\lim _{x \rightarrow 0} \frac{f(x)}{x^3}\) is equal toJEE Mains 2024 Hard
- Let \(g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)\) and \(f^{\prime \prime}(x)>0\) for all \(\mathrm{x} \in(0,3)\). If \(\mathrm{g}\) is decreasing in \((0, \alpha)\) and increasing in \((\alpha, 3)\), then \(8 \alpha\) isJEE Mains 2024 Hard
- Let \( S=\{(m,n): m, n\in\{1,2,3,.....,50\}\} \). If the number of elements (m, n) in S such that \( 6^{m}+9^{n} \) is a multiple of 5 is p and the number of elements (m, n) in S such that \( m+n \) is a square of a prime number is q, then \( p+q \) is equal to :JEE Mains 2026 Medium