Question:
Let \(\theta\) be the acute angle between the tangents to the ellipse \(\frac{x^2}{9}+\frac{y^2}{1}=1\) and the circle \(x^2+y^2=3\) at their point of intersection in the first quadrant. Then \(\tan\theta\) is equal to :
Let \(\theta\) be the acute angle between the tangents to the ellipse \(\frac{x^2}{9}+\frac{y^2}{1}=1\) and the circle \(x^2+y^2=3\) at their point of intersection in the first quadrant. Then \(\tan\theta\) is equal to :
Show Hint
Intersect the curves to get \(\left(\frac32,\frac{\sqrt3}{2}\right)\), find the two tangent slopes by differentiation, then use \(\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|\).
Updated On: Nov 4, 2026
- \(\frac{5}{2\sqrt3}\)
- \(\frac{2}{\sqrt3}\)
- \(\frac{4}{\sqrt3}\)
- \(2\)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Substituting \(y^2=3-x^2\) in the ellipse: \(\frac{x^2}{9}+3-x^2=1\Rightarrow\frac{8x^2}{9}=2\Rightarrow x^2=\frac94\). First quadrant: \(x=\frac32,\ y=\frac{\sqrt3}{2}\).
Step 2: Ellipse: \(\frac{2x}{9}+2y\,y'=0\Rightarrow m_1=-\frac{x}{9y}=-\frac{3/2}{9\cdot\sqrt3/2}=-\frac{1}{3\sqrt3}\).
Step 3: Circle: \(m_2=-\frac xy=-\frac{3/2}{\sqrt3/2}=-\sqrt3\).
Step 4: \(\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|=\left|\frac{-\frac{1}{3\sqrt3}+\sqrt3}{1+\frac13}\right|=\frac{\frac{8}{3\sqrt3}}{\frac43}=\frac{2}{\sqrt3}\).
Answer: (2) \(\frac{2}{\sqrt3}\)
Step 2: Ellipse: \(\frac{2x}{9}+2y\,y'=0\Rightarrow m_1=-\frac{x}{9y}=-\frac{3/2}{9\cdot\sqrt3/2}=-\frac{1}{3\sqrt3}\).
Step 3: Circle: \(m_2=-\frac xy=-\frac{3/2}{\sqrt3/2}=-\sqrt3\).
Step 4: \(\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|=\left|\frac{-\frac{1}{3\sqrt3}+\sqrt3}{1+\frac13}\right|=\frac{\frac{8}{3\sqrt3}}{\frac43}=\frac{2}{\sqrt3}\).
Answer: (2) \(\frac{2}{\sqrt3}\)
Was this answer helpful?
0
0
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
View More Questions
Top JEE Main Conic sections Questions
- Let l be a line which is normal to the curve \(y = 2x^2 + x + 2\) at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to _________ .
- Let the latus rectum of the hyperbola \( \frac{x^2}{9} - \frac{y^2}{b^2} = 1 \) subtend an angle of \( \frac{\pi}{3} \) at the center of the hyperbola. If \( b^2 \) is equal to \( \frac{1}{m} \left( 1 + \sqrt{n} \right) \), where \( l \) and \( m \) are coprime numbers, then \( l^2 + m^2 + n^2 \) is equal to \(\_\_\_\_\_\_\_\_\_\).
- Let \( e_1 \) be the eccentricity of the hyperbola $$ \frac{x^2}{16} - \frac{y^2}{9} = 1 $$ and \( e_2 \) be the eccentricity of the ellipse $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b, $$ which passes through the foci of the hyperbola. If \( e_1 e_2 = 1 \), then the length of the chord of the ellipse parallel to the x-axis and passing through (0, 2) is:
- For \(0 < \theta < \frac{pi}{2}\), if the eccentricity of the hyperbola \( x^2 - y^2 \csc^2 \theta = 5 \) is \( \sqrt{7} \) times the eccentricity of the ellipse \( x^2 \csc^2 \theta + y^2 = 5 \), then the value of \( \theta \) is:
- The area (in square units) of the region enclosed by the ellipse $$ x^2 + 3y^2 = 18 $$ in the first quadrant below the line \( y = x \) is:
View More Questions
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
View More Questions