JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let the parabola \(y = x^2 + px + q\) passing through the point \((1, -1)\) be such that the distance between its vertex and the \(x\)-axis is minimum. Then the value of \(p^2 + q^2\) is:
- A \(2\)
- B \(4\)
- C \(5\)
- D \(8\)
Answer & Solution
Correct Answer
(B) \(4\)
Step-by-step Solution
Detailed explanation
Given the equation of the parabola \(y = x^2 + px + q\). Since it passes through \((1, -1)\), we have: \(-1 = 1 + p + q \Rightarrow q = -p - 2\) The vertex of the parabola \(y = x^2 + px + q\) occurs at \(x = -\dfrac{p}{2}\). The \(y\)-coordinate of the vertex is:…
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 lines \(x -2y = 12\) is tangent to the ellipse \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\) at the point \(\left( {3,\frac{-9}{2}} \right)\), then the length of the latus rectum of the ellipse isJEE Mains 2019 Hard
- If an equation of a tangent to the curve, \(y = \cos \,\left( {x + f} \right),\, - 1\, - \pi \le x \le 1 + \pi ,\) is \(x + 2y = k\) then \(k\) is equal toJEE Mains 2013 Hard
- A plane which bisects the angle between the two given planes \(2x -y + 2z -4 = 0\) and \(x + 2y + 2z -2 = 0\), passes through the pointJEE Mains 2019 Hard
- Let in a Binomial distribution, consisting of \(5\) independent trials, probabilities of exactly \(1\) and \(2\) successes be \(0.4096\) and \(0.2048\) respectively. Then the probability of getting exactly \(3\) successes is equal to ....... .JEE Mains 2021 Hard
- The tangent to the curve, \(y = xe^{x^2}\) passing, through the point \((1, e)\) also passes through the pointJEE Mains 2019 Hard
- \(\lim _{n \rightarrow \infty} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\right\}\) is equal toJEE Mains 2023 Hard
More PYQs from JEE Mains
- The value of the integral \(\int \limits_1^2\left(\frac{t^4+1}{t^6+1}\right) d t\) is \(..........\).JEE Mains 2023 Hard
- If a line, \(y=m x+c\) is a tangent to the circle, \((x-3)^{2}+y^{2}=1\) and it is perpendicular to a line \(\mathrm{L}_{1},\) where \(\mathrm{L}_{1}\) is the tangent to the circle, \(\mathrm{x}^{2}+\mathrm{y}^{2}=1\) at the point \(\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right),\) thenJEE Mains 2020 Hard
- \(\smallint \frac{{2{x^{12}} + 5{x^9}}}{{{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}dx = \)JEE Mains 2016 Hard
- Let \(\frac{1}{{{x_1}}},\frac{1}{{{x_2}}},\frac{1}{{{x_3}}},.....,\) \(({x_i} \ne \,0\,for\,\,i\, = 1,2,....,n)\) be in \(A.P.\) such that \(x_1 = 4\) and \(x_{21} = 20.\) If \(n\) is the least positive integer for which \(x_n > 50,\) then \(\sum\limits_{i = 1}^n {\left( {\frac{1}{{{x_i}}}} \right)} \) is equal to.JEE Mains 2018 Hard
- The tangent to the circle \(C_1 : x^2 + y^2 - 2x- 1\, = 0\) at the point \((2, 1)\) cuts off a chord of length \(4\) from a circle \(C_2\) whose centre is \((3, - 2)\). The radius of \(C_2\) isJEE Mains 2018 Hard
- Consider the parabola with vertex \(\left(\frac{1}{2}, \frac{3}{4}\right)\) and the directrix \(\mathrm{y}=\frac{1}{2}\). Let \(\mathrm{P}\) be the point where the parabola meets the line \(\mathrm{x}=-\frac{1}{2}\). If the normal to the parabola at \(\mathrm{P}\) intersects the parabola again at the point \(\mathrm{Q}\), then \((\mathrm{PQ})^{2}\) is equal to :JEE Mains 2021 Hard