JEE Advanced · Mathematics · 15. Hyperbola
The line \(2 x+y=1\) is tangent to the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\). If this line passes through the point of intersection of the nearest directrix and the \(x\)-axis, then the eccentricity of the hyperbola is
- A 2
- B 4
- C 6
- D 8
Answer & Solution
Correct Answer
(A) 2
Step-by-step Solution
Detailed explanation
On substituting \(\left(\frac{a}{e}, 0\right)\) in \(y=-2 x+1\), we get
\[
\begin{aligned}
0 & =-\frac{2 a}{e}+1 \\
\Rightarrow \quad \frac{a}{e} & =\frac{1}{2}
\end{aligned}
\]
Also, \(y=-2 x+1\) is tangent to hyperbola
\[
\begin{array}{ll}
\therefore & 1=4 a^2-b^2 \\
\Rightarrow & \frac{1}{a^2}=4-\left(e^2-1\right)
\end{array}
\]
\[
\begin{array}{ll}
\Rightarrow & \frac{4}{e^2}=5-e^2 \\
\Rightarrow & e^4-5 e^2+4=0 \\
\Rightarrow & \left(e^2-4\right)\left(e^2-1\right)=0 \\
\Rightarrow & e=2, e=1
\end{array}
\]
\(e=1\) gives the conic as parabola. But conic is given as hyperbola, hence \(e=2\)
\[
\begin{aligned}
0 & =-\frac{2 a}{e}+1 \\
\Rightarrow \quad \frac{a}{e} & =\frac{1}{2}
\end{aligned}
\]
Also, \(y=-2 x+1\) is tangent to hyperbola
\[
\begin{array}{ll}
\therefore & 1=4 a^2-b^2 \\
\Rightarrow & \frac{1}{a^2}=4-\left(e^2-1\right)
\end{array}
\]
\[
\begin{array}{ll}
\Rightarrow & \frac{4}{e^2}=5-e^2 \\
\Rightarrow & e^4-5 e^2+4=0 \\
\Rightarrow & \left(e^2-4\right)\left(e^2-1\right)=0 \\
\Rightarrow & e=2, e=1
\end{array}
\]
\(e=1\) gives the conic as parabola. But conic is given as hyperbola, hence \(e=2\)
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 Mathematics
- Let be a function which is continuous on and is differentiable on . Let for If for all then equalsJEE Advanced 2014 Easy
- Paragraph:
Consider the polynomial \(f(x)=1+2 x+3 x^2+4 x^3\). Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t=|s|\).Question:
The function \(f^{\prime}(x)\) isJEE Advanced 2010 Medium - Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow(0,4)\) be functions defined by \(f(x)=\log _e\left(x^2+2 x+4\right)\), and \(g(x)=\frac{4}{1+e^{-2 x}}\)
Define the composite function \(f \circ g^{-1}\) by \(\left(f \circ g^{-1}\right)(x)=\left(g^{-1}(x)\right)\), where \(g^{-1}\) is the inverse of the function \(g\).
Then the value of the derivative of the composite function \(f \circ g^{-1}\) at \(x=2\) is ________ .JEE Advanced 2025 Hard - Let be a point in the first octant, whose image in the plane (that is, the line segment is perpendicular to the plane and the mid-point of lies in the plane ) lies on the z-axis. Let the distance of from the x-axis be . If is the image of in the xy-plane, then the length of is ______.JEE Advanced 2018 Medium
- Let \(m\) be the smallest positive integer such that the coefficient of \(x^2\) in the expansion of \((1+x)^2 +(1+x)^3+\ldots+(1+x)^{49}\) \(+~(1+m x)^{50}\) is \((3 n+1){ }^{51} C_3\) for some positive integer n . Then the value of \(n\) isJEE Advanced 2016 Medium
- Paragraph:
Let \(A\) be the set of all \(3 \times 3\) symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0.
Question:
The number of matrices in \(A\) isJEE Advanced 2009 Medium
More PYQs from JEE Advanced
- \[
\text { Let } M \text { be a } 3 \times 3 \text { matrix satisfying }
\]
\[
M\left[\begin{array}{l}
0 \\
1 \\
0
\end{array}\right]=\left[\begin{array}{c}
-1 \\
2 \\
3
\end{array}\right], M\left[\begin{array}{c}
1 \\
-1 \\
0
\end{array}\right]=\left[\begin{array}{c}
1 \\
1 \\
-1
\end{array}\right] \text { and } M\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right]=\left[\begin{array}{c}
0 \\
0 \\
12
\end{array}\right]
\]
Then, the sum of the diagonal entries of \(M\) isJEE Advanced 2011 Easy - A large square container with thin transparent vertical walls and filled with water is kept on a horizontal table. A student holds a thin straight wire vertically inside the water from one of its corners, as shown schematically in the figure. Looking at the wire from this corner, another student sees two images of the wire, located symmetrically on each side of the line of sight as shown. The separation (in ) between these images is ________.
JEE Advanced 2020 Hard - Paragraph:
Consider the function \(f:(-\infty, \infty) \rightarrow(-\infty, \infty)\) defined by \(f(x)=\frac{x^2-a x+1}{x^2+a x+1} ; 0 < a < 2\)Question:
Which of the following is true?JEE Advanced 2008 Easy - Two transparent media of refractive indices \(\mu_1\) and \(\mu_3\) have a solid lens shaped transparent material of refractive index \(\mu_2\) between them as shown in figures in Column II. A ray traversing these media is also shown in the figures. In Column I different relationships between \(\mu_1, \mu_2\) and \(\mu_3\) are given. Match them to the ray diagram shown in Column II.
JEE Advanced 2010 Medium - The compound(s) with two lone pairs of electrons on the central atom is (are)JEE Advanced 2016 Medium
- Considering the reaction sequence given below, the correct statement(s) is(are)
JEE Advanced 2022 Medium