Question:
Consider the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ with foci at $S$ and $S_1$, where $S$ lies on the positive $x$-axis Let $P$ be a point on the hyperbola, in the first quadrant Let $\angle \operatorname{SPS}_1=\alpha$, with $\alpha<\frac{\pi}{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S _1 P$ at $P _1$ Let $\delta$ be the distance of $P$ from the straight line $SP _1$, and $\beta= S _1 P$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is _____.
Consider the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ with foci at $S$ and $S_1$, where $S$ lies on the positive $x$-axis Let $P$ be a point on the hyperbola, in the first quadrant Let $\angle \operatorname{SPS}_1=\alpha$, with $\alpha<\frac{\pi}{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S _1 P$ at $P _1$ Let $\delta$ be the distance of $P$ from the straight line $SP _1$, and $\beta= S _1 P$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is _____.
Updated On: Apr 23, 2024
Hide Solution
Verified By Collegedunia
Correct Answer: 7
Solution and Explanation
The answer is 7.
Was this answer helpful?
0
0
Top Questions on Conic sections
- The roots of the quadratic equation \( x^2 - 16 = 0 \) are:
- TS POLYCET - 2025
- Mathematics
- Conic sections
- If $ \theta $ is the angle subtended by a latus rectum at the center of the hyperbola having eccentricity $$ \frac{2}{\sqrt{7} - \sqrt{3}}, $$ then find $ \sin \theta $.
- AP EAPCET - 2025
- Mathematics
- Conic sections
- The tangent drawn at an extremity (in the first quadrant) of latus rectum of the hyperbola $$ \frac{x^2}{4} - \frac{y^2}{5} = 1 $$ meets the x-axis and y-axis at $ A $ and $ B $ respectively. If $ O $ is the origin, find $$ (OA)^2 - (OB)^2. $$
- AP EAPCET - 2025
- Mathematics
- Conic sections
- If the normal drawn at the point $$ P \left(\frac{\pi}{4}\right) $$ on the ellipse $$ x^2 + 4y^2 - 4 = 0 $$ meets the ellipse again at $ Q(\alpha, \beta) $, then find $ \alpha $.
- AP EAPCET - 2025
- Mathematics
- Conic sections
- If $ x - y - 3 = 0 $ is a normal drawn through the point $ (5, 2) $ to the parabola $ y^2 = 4x $, then the slope of the other normal that can be drawn through the same point to the parabola is?
- AP EAPCET - 2025
- Mathematics
- Conic sections
View More Questions
Questions Asked in JEE Advanced exam
- A projectile is thrown at an angle of \(60^\circ\) with the horizontal. Initial speed is \(270\, \text{m/s}\). A linear drag force \(F = -CV\) acts on the body. Find the horizontal displacement till \(t = 2\) seconds. Given \(C = 0.1\, \text{s}^{-1}\).
- JEE Advanced - 2025
- Projectile motion
- A single slit diffraction experiment is performed to determine the slit width using the equation, $ \dfrac{b d}{D} = m \lambda $, where $ b $ is the slit width, $ D $ the shortest distance between the slit and the screen, $ d $ the distance between the $ m^{\text{th}} $ diffraction maximum and the central maximum, and $ \lambda $ is the wavelength. $ D $ and $ d $ are measured with scales of least count of 1 cm and 1 mm, respectively. The values of $ \lambda $ and $ m $ are known precisely to be $ 600 \, \text{nm} $ and 3, respectively. The absolute error (in $ \mu\text{m} $) in the value of $ b $ estimated using the diffraction maximum that occurs for $ m = 3 $ with $ d = 5 \, \text{mm} $ and $ D = 1 \, \text{m} $ is ___.
- JEE Advanced - 2025
- Error Propagation
- Let $$ \alpha = \frac{1}{\sin 60^\circ \sin 61^\circ} + \frac{1}{\sin 62^\circ \sin 63^\circ} + \cdots + \frac{1}{\sin 118^\circ \sin 119^\circ}. $$ Then the value of $$ \left( \frac{\csc 1^\circ}{\alpha} \right)^2 $$ is \rule{1cm}{0.15mm}.
- JEE Advanced - 2025
- Some Applications of Trigonometry
Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter $ D $ of a tube. The measured value of $ D $ is:
- JEE Advanced - 2025
- Units and measurement
- Let $ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $ and $ P = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} $. Let $ Q = \begin{pmatrix} x & y \\ z & 4 \end{pmatrix} $ for some non-zero real numbers $ x, y, z $, for which there is a $ 2 \times 2 $ matrix $ R $ with all entries being non-zero real numbers, such that $$ QR = RP $$ Then which of the following statements is (are) TRUE?
- JEE Advanced - 2025
- Matrices
View More Questions
Concepts Used:
Hyperbola
Hyperbola is the locus of all the points in a plane such that the difference in their distances from two fixed points in the plane is constant.
Hyperbola is made up of two similar curves that resemble a parabola. Hyperbola has two fixed points which can be shown in the picture, are known as foci or focus. When we join the foci or focus using a line segment then its midpoint gives us centre. Hence, this line segment is known as the transverse axis.
