Question:
If the line $2x+\sqrt6y=2$ touches the hyperbola $x^2-2y^2=4$, then the point of contact is
If the line $2x+\sqrt6y=2$ touches the hyperbola $x^2-2y^2=4$, then the point of contact is
Updated On: Jun 14, 2022
- $(-2,\sqrt6)$
- $(-5,2\sqrt6)$
- $\Bigg(\frac{1}{2},\frac{1}{\sqrt6}\Bigg)$
- $(4,-\sqrt6)$
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
The equation of tangent at $(x_1,y_1)$ is $xx_1-2yy_1=4$,which is same as $2x+\sqrt6y=2.$
$\therefore\, \, \, \, \, \, \, \, \frac{x_1}{2}=-\frac{2y_1}{\sqrt6}=\frac{4}{2}$
$\Rightarrow\, \, \, \, \, \, \, \, \, x_1=4$ and $y_1=-\sqrt6$
Thus, the point of contact is $(4,-\sqrt6)$.
$\therefore\, \, \, \, \, \, \, \, \frac{x_1}{2}=-\frac{2y_1}{\sqrt6}=\frac{4}{2}$
$\Rightarrow\, \, \, \, \, \, \, \, \, x_1=4$ and $y_1=-\sqrt6$
Thus, the point of contact is $(4,-\sqrt6)$.
Was this answer helpful?
0
0
Top Questions on Conic sections
- Let \[ E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b \quad \text{and} \quad H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1. \] Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to:
- JEE Main - 2025
- Mathematics
- Conic sections
- Let \[ E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b \quad \text{and} \quad H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1. \] Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to:
- JEE Main - 2025
- Mathematics
- Conic sections
- The area bounded by \( y - 1 = |x| \) and \( y + 1 = |x| \) is: (a) \( \frac{1}{2} \)
- VITEEE - 2024
- Mathematics
- Conic sections
If a tangent to the hyperbola \( x^2 - \frac{y^2}{3} = 1 \) is also a tangent to the parabola \( y^2 = 8x \), then the equation of such tangent with the positive slope is:
- AP EAPCET - 2024
- Mathematics
- Conic sections
If a circle of radius 4 cm passes through the foci of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and is concentric with the hyperbola, then the eccentricity of the conjugate hyperbola of that hyperbola is:
- AP EAPCET - 2024
- Mathematics
- Conic sections
View More Questions
Questions Asked in JEE Advanced exam
- Let \(S=\left\{\begin{pmatrix} 0 & 1 & c \\ 1 & a & d\\ 1 & b & e \end{pmatrix}:a,b,c,d,e\in\left\{0,1\right\}\ \text{and} |A|\in \left\{-1,1\right\}\right\}\), where |A| denotes the determinant of A. Then the number of elements in S is _______.
- JEE Advanced - 2024
- Matrices
- A positive, singly ionized atom of mass number \( A_M \) is accelerated from rest by the voltage \( 192 \, \text{V} \). Thereafter, it enters a rectangular region of width \( w \) with magnetic field \( \vec{B}_0 = 0.1\hat{k} \, \text{T} \). The ion finally hits a detector at the distance \( x \) below its starting trajectory. Which of the following option(s) is(are) correct? \[ \text{(Given: Mass of neutron/proton = } \frac{5}{3} \times 10^{-27} \, \text{kg, charge of the electron = } 1.6 \times 10^{-19} \, \text{C).} \]
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass \( m \) and radius \( r \) and is in a uniform vertical magnetic field \( B_0 \). When a current \( I \) is passed through the loop, the loop turns about the line PQ by an angle \( \theta \). The angle \( \theta \) is given by:
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A region in the form of an equilateral triangle (in x-y plane) of height \( L \) has a uniform magnetic field \( \mathbf{B} \) pointing in the \( +z \)-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height \( L \), is placed in the x-y plane with its vertex \( P \) at \( x = 0 \) in the orientation shown in the figure. At \( t = 0 \), the loop starts entering the region of the magnetic field with a uniform velocity \( \mathbf{v} \) along the \( +x \)-direction. The plane of the loop and its orientation remain unchanged throughout its motion.
Which of the following graphs best depicts the variation of the induced emf (\( \mathcal{E} \)) in the loop as a function of the distance (\( x \)) starting from \( x = 0 \)?- JEE Advanced - 2024
- Radioactivity
- A table tennis ball has radius \( \frac{3}{2} \times 10^{-2} \, \text{m} \) and mass \( \frac{22}{7} \times 10^{-3} \, \text{kg} \). It is slowly pushed down into a swimming pool to a depth \( d = 0.7 \, \text{m} \) below the water surface and then released from rest. It emerges from the water surface at speed \( v \), without getting wet, and rises to a height \( H \). Which of the following option(s) is(are) correct?\(\text{(Given: } \pi = \frac{22}{7}, g = 10 \, \text{m/s}^2, \text{density of water} = 1 \times 10^3 \, \text{kg/m}^3, \text{viscosity of water} = 1 \times 10^{-3} \, \text{Pa.s).}\)
- JEE Advanced - 2024
- Radioactivity
View More Questions