enEnglishguગુજરાતી
JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let a tangent be drawn to the ellipse \(\frac{x^{2}}{27}+y^{2}=1\) at \((3 \sqrt{3} \cos \theta, \sin \theta)\) where \(\theta \in\left(0, \frac{\pi}{2}\right)\). Then the value of \(\theta\) such that the sum of intercepts on axes made by this tangent is minimum is equal to ..... .
- A \(\frac{\pi}{8}\)
- B \(\frac{\pi}{4}\)
- C \(\frac{\pi}{6}\)
- D \(\frac{\pi}{3}\)
Answer & Solution
Correct Answer
(C) \(\frac{\pi}{6}\)
Step-by-step Solution
Detailed explanation
Equation of tangent be \(\frac{ x \cos \theta}{3 \sqrt{3}}+\frac{ y \cdot \sin \theta}{1}=1, \quad \theta \in\left(0, \frac{\pi}{2}\right)\) intercept on \(x\) -axis \(OA =3 \sqrt{3} sec \theta\) intercept on \(y-\)axis \(OB =\operatorname{cosec} \theta\) Now, sum of intercept…
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
- If \(\mathrm{n}\) is the number of solutions of the equation \(2 \cos x\left(4 \sin \left(\frac{\pi}{4}+x\right) \sin \left(\frac{\pi}{4}-x\right)-1\right)=1, x \in[0, \pi]\) and \(S\) is the sum of all these solutions, then the ordered pair \((\mathrm{n}, \mathrm{S})\) is :JEE Mains 2021 Hard
- The number of real solutions of \(x^{7}+5 x^{3}+3 x+1=\) \(0\) is equal to............JEE Mains 2022 Medium
- Two circles each of radius \(5\, units\) touch each other at the point \((1,2)\). If the equation of their common tangent is \(4 \mathrm{x}+3 \mathrm{y}=10\), and \(\mathrm{C}_{1}(\alpha, \beta)\) and \(\mathrm{C}_{2}(\gamma, \delta)\), \(\mathrm{C}_{1} \neq \mathrm{C}_{2}\) are their centres, then \(|(\alpha+\beta)(\gamma+\delta)|\) is equal to .... .JEE Mains 2021 Hard
- Let the shortest distance between the lines \(L : \frac{ x -5}{-2}=\frac{ y -\lambda}{0}=\frac{ z +\lambda}{1}, \lambda \geq 0\) and \(L _1: x +1= y -\) \(1=4-z\) be \(2 \sqrt{6}\). If \((\alpha, \beta, \gamma)\) lies on \(L\), then which of the following is NOT possible?JEE Mains 2023 Hard
- The sum of first \(9\) terms of the series \(\frac{{{1^3}}}{1} + \frac{{{1^3} + {2^3}}}{{1 + 3}} + \frac{{{1^3} + {2^3} + {3^3}}}{{1 + 3 + 5}} + .\;.\;.\;.\)JEE Mains 2015 Hard
- A plane passes through the points \(A (1,2,3), B (2,3,1)\) and \(C (2,4,2)\). If \(O\) is the origin and \(P\) is \((2,-1,1) ,\) then the projection of \(\overline{ OP }\) on this plane is of length .... .JEE Mains 2021 Medium
More PYQs from JEE Mains
- The number of ways in which \(21\) identical apples can be distributed among three children such that each child gets at least \(2\) apples, isJEE Mains 2024 Medium
- If the tangent at a point \(P\) on the parabola \(y ^2=3 x\) is parallel to the line \(x+2 y=1\) and the tangents at the points \(Q\) and \(R\) on the ellipse \(\frac{x^2}{4}+\frac{y^2}{1}=1\) are perpendicular to the line \(x-y=2\), then the area of the triangle \(PQR\) is:JEE Mains 2023 Hard
- The relation \(R=\{(x, y): x, y \in \mathbb{Z}\) and \(x+y\) is even \(\}\) is:JEE Mains 2025 Medium
- Let \(S_n\) denote the sum of first \(n\) terms an arithmetic progression. If \(S_{20}=790\) and \(S_{10}=145\), then \(S_{15}-\) \(S_5\) is :JEE Mains 2024 Medium
- Let the plane \(P: \vec{r} \cdot \vec{a}=d\) contain the line of intersection of two planes \(\overrightarrow{ r } \cdot(\hat{ i }+3 \hat{ j }-\hat{ k })=6\) and \(\overrightarrow{ r } \cdot(-6 \hat{ i }+5 \hat{ j }-\hat{ k })=7\). If the plane \(P\) passes through the point \(\left(2,3, \frac{1}{2}\right)\), then the value of \(\frac{|13 \vec{a}|^{2}}{ d ^{2}}\) is equal toJEE Mains 2022 Hard
- If \(0 < x < \frac{1}{\sqrt{2}}\) and \(\frac{\sin ^{-1} x}{\alpha}=\frac{\cos ^{-1} x}{\beta}\), then a value of \(\sin \left(\frac{2 \pi \alpha}{\alpha+\beta}\right)\) is\(......\)JEE Mains 2022 Hard