JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let \(L_1, L_2\) be the lines passing through the point \(\mathrm{P}(0,1)\) and touching the parabola \(9 x^2+12 x+18 y-14=0\). Let \(Q\) and \(R\) be the points on the lines \(\mathrm{L}_1\) and \(\mathrm{L}_2\) such that the \(\triangle \mathrm{PQR}\) is an isosceles triangle with base \(\mathrm{QR}\). If the slopes of the lines \(Q R\) are \(m_1\) and \(m_2\). then \(16\left(m_1^2+m_2^2\right)\) is equal to ..............
- A \(68\)
- B \(25\)
- C \(46\)
- D \(74\)
Answer & Solution
Correct Answer
(A) \(68\)
Step-by-step Solution
Detailed explanation
\( 9 x^2+12 x+4=-18(y-1) \) \( (3 x+2)^2=-18(y-1) \) \( \left(x+\frac{2}{3}\right)^2=-2(y-1)\) \( (0,1) \) \( y=m x+1 \) \( \left(x+\frac{2}{3}\right)^2=-2(y-1) \) \( (3 x+2)^2=-18 m x \) \( 9 x^2+(12+18 m) x+4=0 \) \( 4(6+9 m)^2=4(36) \) \( 6+9 m=6,-6 \) \( m=0, \frac{-4}{3}\)…
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 the coefficients of the three successive terms in the binomial expansion of \((1 + x)^n\) are in the ratio \(1 : 7 : 42,\) then the first of these terms in the expansion isJEE Mains 2015 Hard
- Let \(a\) and \(\mathrm{b}\) respectively be the points of local maximum and local minimum of the function \(f(x)=2 x^{3}-3 x^{2}-12 x .\) If \(A\) is the total area of the region bounded by \(\mathrm{y}=\mathrm{f}(\mathrm{x})\), the \(\mathrm{x}\)-axis and the lines \(x=a\) and \(x=b\), then \(4 A\) is equal to ..... .JEE Mains 2021 Hard
- If \(\lim _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{m \sqrt{5}}{n(2 n)^{2 / 3}}\), where \(\operatorname{gcd}(m, n)=1\), then \(8 m+12 n\) is equal to ...........JEE Mains 2024 Hard
- If the points of intersection of the ellipses \( x^{2}+2y^{2}-6x-12y+23=0 \) and \( 4x^{2}+2y^{2}-20x-12y+35=0 \) lie on a circle of radius \( r \) and centre \( (a, b) \), then the value of \( ab+18r^{2} \) is:JEE Mains 2026 Easy
- The function \(f(x)=\left|x^{2}-2 x-3\right| \cdot e^{\left|9 x^{2}-12 x+4\right|}\) is not differentiable at exactly :JEE Mains 2021 Hard
- If \(\sum_{r=1}^{50} \tan ^{-1} \frac{1}{2 r^{2}}=p\), then the value of \(\tan p\) is :JEE Mains 2021 Hard
More PYQs from JEE Mains
- Let \(K\) be the sum of the coefficients of the odd powers of \(x\) in the expansion of \((1+ x )^{99}\). Let a be the middle term in the expansion of \(\left(2+\frac{1}{\sqrt{2}}\right)^{200}\). If \(\frac{{ }^{200} C _{99} K }{ a }=\frac{2^{\ell} m }{ n }\), where \(m\) and \(n\) are odd numbers, then the ordered pair \((l, n )\) is equal to :JEE Mains 2023 Hard
- Let \(f ( x )\) be a polynomial of degree \(6\) in \(x ,\) in which the coefficient of \(x^{6}\) is unity and it has extrema at \(x=-1\) and \(x=1\). If \(\lim _{x \rightarrow 0} \frac{f(x)}{x^{3}}=1,\) then \(5 \cdot f (2)\) is equal to .............JEE Mains 2021 Hard
- Total numbers of \(3-\)digit numbers that are divisible by 6 and can be formed by using the digits \(1, 2, 3, 4,5\) with repetition, is \(.......\).JEE Mains 2023 Hard
- The set of all values of \(\lambda\) for which the system of linear \(2{x_1} - 2{x_2} + {x_3} = \lambda {x_1}\;,\;2{x_1} - 3{x_2} + 2{x_3} = \lambda {x_2}\;\;,\)\(\;\; - {x_1} + 2{x_2} = \lambda {x_3}\) has a non-trivial solutionJEE Mains 2015 Medium
- Let \(z\) be a complex number such that \(|z+2| = |z-2|\) and \(\arg\left(\dfrac{z+3}{z-i}\right) = \dfrac{\pi}{4}\). Then \(|z|^2\) is equal to:JEE Mains 2026 Medium
- If the sum of the deviations of \(50\) observations from \(30\) is \(50\), then the mean of these observations isJEE Mains 2019 Hard