JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let the common tangents to the curves \(4\left(x^{2}+y^{2}\right)=\) \(9\) and \(y ^{2}=4 x\) intersect at the point \(Q\). Let an ellipse, centered at the origin \(O\), has lengths of semi-minor and semi-major axes equal to \(OQ\) and \(6\) , respectively. If \(e\) and \(l\) respectively denote the eccentricity and the length of the latus rectum of this ellipse, then \(\frac{l}{ e ^{2}}\) is equal to
- A \(5\)
- B \(4\)
- C \(3\)
- D \(2\)
Answer & Solution
Correct Answer
(B) \(4\)
Step-by-step Solution
Detailed explanation
\(x^{2}+y^{2}=\frac{9}{4} \quad y=4 x\) Equation tangent in slope form \(y=m x \pm \frac{3}{2} \sqrt{\left(1+m^{2}\right)}\) .....\((1)\) \(y=m x+\frac{1}{m}\) .....\((2)\) compare \((1)\) and \((2)\) \(\pm \frac{3}{2} \sqrt{\left(1+m^{2}\right)}=\frac{1}{m^{2}}\)…
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 Maths
- If the arithmetic mean of two numbers \(a\) and \(b, a>b>0\), is five times their geometric mean, then \(\frac{{a + b}}{{a - b}}\) is equal toJEE Mains 2017 Hard
- Let \(f\) be a differentiable function satisfying \(f ( x )=\frac{2}{\sqrt{3}} \int_{0}^{\sqrt{3}} f \left(\frac{\lambda^{2} x }{3}\right) d \lambda, x >0\) and \(f (1)=\sqrt{3}\). If \(y=f(x)\) passes through the point \((\alpha, 6)\), then \(\alpha\) is equal to \(.........\)JEE Mains 2022 Hard
- Consider a region \(\mathrm{R}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathrm{R}^{2}: \mathrm{x}^{2} \leq \mathrm{y} \leq 2 \mathrm{x}\right\}\) If a line \(\mathrm{y}=\alpha\) divides the area of region \(\mathrm{R}\) into two equal parts, then which of the following is true?JEE Mains 2020 Hard
- The number of all possible positive integral values of \(\alpha \) for which the roots of the quadratic equation, \(6x^2 - 11x +\alpha =0\) are rational numbers isJEE Mains 2019 Hard
- Let the plane containing the line of intersection of the planes \(P 1: x+(\lambda+4) y+z=1\) and \(P2:\) \(2 x+y+z=2\) pass through the points \((0,1,0)\) and \((1,0,1)\). Then the distance of the point \((2 \lambda, \lambda,-\lambda)\) from the plane \(P 2\) isJEE Mains 2023 Hard
- If \(A\) and \(B\) are two events such that \(P(A \cap B)=0.1\), and \(P(A \mid B)\) and \(P(B \mid A)\) are the roots of the equation \(12 x^2-7 x+1=0\), then the value of \(\frac{\mathrm{P}(\overline{\mathrm{A}} \cup \overline{\mathrm{B}})}{\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})}\) is :JEE Mains 2025 Hard
More PYQs from JEE Mains
- The locus of the foot of perpendicular drawn from the centre of the ellipse \({x^2} + 3{y^2} = 6\) on any tangent to it isJEE Mains 2014 Hard
- Let \(\mathrm{y}=\mathrm{y}(\mathrm{x})\) be a solution of the differential equation, \(\sqrt{1-\mathrm{x}^{2}} \frac{\mathrm{dy}}{\mathrm{dx}}+\sqrt{1-\mathrm{y}^{2}}=0,|\mathrm{x}|<1\) If \(\mathrm{y}\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2},\) then \(\mathrm{y}\left(\frac{-1}{\sqrt{2}}\right)\) is equal toJEE Mains 2020 Hard
- The equation of a circle is \(\operatorname{Re}\left(z^{2}\right)+2(\operatorname{Im}(z))^{2}+2 \operatorname{Re}(z)=0\), where \(z=x+\) iy. A line which passes through the center of the given circle and the vertex of the parabola, \(x^{2}-6 x-y+13=0,\) has \(y\)-intercept equal to \(.....\)JEE Mains 2021 Hard
- lf the mean deviation of the numbers \(1, 1 + d, . . . ,1 + 100d\) from their mean is \(255,\) then a value of \(d\) isJEE Mains 2016 Hard
- Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by
\(f(x)=(2+3 a) x^2+\left(\frac{a+2}{a-1}\right) x+b, a \neq 1 . \text { If }\)
\(f(x+\mathrm{y})=f(x)+f(\mathrm{y})+1-\frac{2}{7} x \mathrm{y}\), then the value of \(28 \sum_{i=1}^5|f(i)|\) isJEE Mains 2025 Medium - Let \(f : R \rightarrow R\) be a function such that \(f(x)=\frac{x^2+2 x+1}{x^2+1}\). ThenJEE Mains 2023 Hard