JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let \(\mathrm{P}\) be a parabola with vertex \((2,3)\) and directrix \(2 x+y=6\). Let an ellipse \(E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b\) of eccentricity \(\frac{1}{\sqrt{2}}\) pass through the focus of the parabola \(\mathrm{P}\). Then the square of the length of the latus rectum of \(\mathrm{E}\), is
- A \(\frac{385}{8}\)
- B \(\frac{347}{8}\)
- C \(\frac{512}{25}\)
- D \(\frac{656}{25}\)
Answer & Solution
Correct Answer
(D) \(\frac{656}{25}\)
Step-by-step Solution
Detailed explanation
\(\text {Slope of axis }=\frac{1}{2}\) \(y-3=\frac{1}{2}(x-2)\) \(\Rightarrow 2 y-6=x-2\) \(\Rightarrow 2 y-x-4=0\) \(2 x+y-6=0\) \(4 x+2 y-12=0\) \(\alpha+1.6=4 \Rightarrow \alpha=2.4\) \(\beta+2.8=6 \Rightarrow \beta=3.2\) Ellipse passes through \((2.4,3.2)\)…
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 \(y=y(x)\) be the solution of the differential equation \(x\frac{dy}{dx}-y=x^{2}\cot x, x\in(0,\pi)\). If \(y(\frac{\pi}{2})=\frac{\pi}{2}\), then \(6y(\frac{\pi}{6})-8y(\frac{\pi}{4})\) is equal to :JEE Mains 2026 Easy
- Let \(S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^2 x}+9^{\tan ^2 x}=10\right\}\) and \(\beta=\sum_{x \in S} \tan ^2\left(\frac{x}{3}\right)\), then \(\frac{1}{6}(\beta-14)^2\) is equal toJEE Mains 2023 Hard
- If \(\alpha \), \(\beta \) and \(\gamma \) are three consecutive terms of a non-constant \(G.P.\) such that the equations \(\alpha x^2 + 2\beta x + \gamma = 0\) and \(x^2 + x -1 = 0\) have a common root, then \(\alpha(\beta + \gamma )\) is equal toJEE Mains 2019 Hard
- A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point A, with uniform speed. At that point, angle of depression of the boat with the man's eye is \(30^{\circ}\) (Ignore man's height). After sailing for \(20\) seconds, towards the base of the tower (which is at the level of water), the boat has reached a point \(B\), where the angle of depression is \(45^{\circ}\). Then the time taken (in seconds) by the boat from \(B\) to reach the base of the tower isJEE Mains 2021 Hard
- A bird is sitting on the top of a vertical pole \(20\, m\) high and its elevation from a point \(O\) on the ground is \(45^o \) . It flies off horizontally straight away from the point \(O\). After one second, the elevation of the bird from \(O\) is reduced to \(30^o \) . Then the speed (in \(m/s\)) of the bird isJEE Mains 2014 Hard
- Let the lines \((2-i) z=(2+i) \bar{z}\) and \((2+i) z+(i-2) \bar{z}-4 i=0,\) (here \(\left.i^{2}=-1\right)\) be normal to a circle \(C\). If the line \(iz +\overline{ z }+1+ i =0\) is tangent to this circle \(C\), then its radius isJEE Mains 2021 Hard
More PYQs from JEE Mains
- The equation of a plane containing the line of intersection of the planes \(2x - y - 4 = 0\) and \(y + 2z - 4 = 0\) and passing through the point \((1, 1, 0)\) isJEE Mains 2019 Medium
- Let \(y=y(x)\) be the solution curve of the differential equation secy \(\frac{d y}{d x}+2 x \sin y=x^3 \cos y\), \(y(1)=0\). Then \(y(\sqrt{3})\) is equal to :JEE Mains 2024 Hard
- For \(k \in N\), if the sum of the series \(1+\frac{4}{ k }+\frac{8}{ k ^2}+\frac{13}{ k ^3}+\frac{19}{ k ^4}+\ldots\) is 10 , then the value of \(k\) isJEE Mains 2023 Hard
- The value of \(lim_{x\rightarrow0}\frac{log_{e}(sec(ex) \cdot sec(e^{2}x)\cdot...\cdot sec(e^{10}x))}{e^{2}-e^{2cos~x}}\) is equal toJEE Mains 2026 Hard
- The system of linear equations \(3 x-2 y-k z=10\); \(2 x-4 y-2 z=6\) ; \(x+2 y-z=5\, m\) is inconsistent ifJEE Mains 2021 Medium
- Let \(\alpha\) and \(\beta\) be real numbers. Consider a \(3 \times 3\) matrix \(A\) such that \(A ^2=3 A +\alpha I\). If \(A ^4=21 A +\beta I\), thenJEE Mains 2023 Hard