JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(\mathrm{P}(\mathrm{h}, \mathrm{k})\) be a point on the curve \(\mathrm{y}=\mathrm{x}^{2}+7 \mathrm{x}+2\) nearest to the line, \(y=3 x-3 .\) Then the equation of the normal to the curve at \(\mathrm{P}\) is
- A \(x+3 y-62=0\)
- B \(x-3 y-11=0\)
- C \(x-3 y+22=0\)
- D \(x+3 y+26=0\)
Answer & Solution
Correct Answer
(D) \(x+3 y+26=0\)
Step-by-step Solution
Detailed explanation
Let \(\mathrm{L}\) be the common normal to parabola \(y=x^{2}+7 x+2\) and line \(y=3 x-3\) \(\Rightarrow\) slope of tangent of \(y=x^{2}+7 x+2\) at \(P=3\) \(\left.\Rightarrow \frac{\mathrm{d} \mathrm{y}}{\mathrm{dx}}\right]_{\mathrm{For} \mathrm{P}}=3\)…
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
- Let the domain of the function
\(f(x)=\cos ^{-1}\left(\frac{4 x+5}{3 x-7}\right)\) be \([\alpha, \beta]\) and the domain of \(\mathrm{g}(\mathrm{x})=\log _2\left(2-6 \log _{27}(2 \mathrm{x}+5)\right)\) be \((\gamma, \delta)\).
Then \(|7(\alpha+\beta)+4(\gamma+\delta)|\) is equal to ________JEE Mains 2025 Medium - If the system of equations \(2 x+y-z=5\) \(2 x-5 y+\lambda z=\mu\) \(x+2 y-5 z=7\) has infinitely many solutions, then \((\lambda+\mu)^2+(\lambda-\mu)^2\) is equal toJEE Mains 2023 Hard
- All the pairs \((x, y)\) that satisfy the inequality \({2^{\sqrt {{{\sin }^2}{\kern 1pt} x - 2\sin {\kern 1pt} x + 5} }}.\frac{1}{{{4^{{{\sin }^2}\,y}}}} \leq 1\) also Satisfy the equationJEE Mains 2019 Hard
- Let the tangent and normal at the point \((3 \sqrt{3}, 1)\) on the ellipse \(\frac{x^2}{36}+\frac{y^2}{4}=1\) meet the \(y\)-axis at the points \(A\) and \(B\) respectively. Let the circle \(C\) be drawn taking \(A B\) as a diameter and the line \(x =2 \sqrt{5}\) intersect \(C\) at the points \(P\) and \(Q\). If the tangents at the points \(P\) and \(Q\) on the circle intersect at the point \((\alpha, \beta)\), then \(\alpha^2-\beta^2\) is equal toJEE Mains 2023 Hard
- If \(\left| {\begin{array}{*{20}{c}}
{{a^2}}&{{b^2}}&{{c^2}} \\
{{{(a + \lambda )}^2}}&{{{(b + \lambda )}^2}}&{{{(c + \lambda )}^2}} \\
{{{(a - \lambda )}^2}}&{{{(b - \lambda )}^2}}&{{{(c - \lambda )}^2}}
\end{array}} \right|\) \( = \,k\lambda \,\,\left| {{\mkern 1mu} {\mkern 1mu} \begin{array}{*{20}{c}}
{{a^2}}&{{b^2}}&{{c^2}} \\
a&b&c \\
1&1&1
\end{array}} \right|,\lambda \, \ne \,0\) then \(k\) is equal toJEE Mains 2014 Hard - The sum of the series : \((2)^2 + 2(4)^2 + 3(6)^2 + ...\) upto \(10\) terms isJEE Mains 2013 Medium
More PYQs from JEE Mains
- If the point \(\left( {2,\alpha ,\beta } \right)\) lies on the plane which passes through the points \((3, 4, 2)\) and \((7, 0, 6)\) and is perpendicular to the plane \(2x - 5y = 15\) , then is equal to \({2\alpha - 3\beta }\) is equal toJEE Mains 2019 Hard
- Consider two sets \(A =\{ x \in z :|(| x -3|-3)| \leq 1\}\) and \(B=\left\{x \in R -\{1,2\}: \frac{(x-2)(x-4)}{x-1} \log _e(|x-2|)=0\right\}\). Then the number of onto functions \( f:A\rightarrow B \) is equal to:JEE Mains 2026 Easy
- The set of all real values of \(\lambda\) for which the function \(f(x)=\left(1-\cos ^{2} x\right) \cdot(\lambda+\sin x)\) \(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right),\) has exactly one maxima and exactly one minima, isJEE Mains 2020 Hard
- Let \(\alpha=\frac{-1+i\sqrt{3}}{2}\) and \(\beta=\frac{-1-i\sqrt{3}}{2}\),\(i=\sqrt{-1}\). If \((7-7\alpha+9\beta)^{20}+(9+7\alpha-7\beta)^{20}+(-7+9\alpha+7\beta)^{20}+(14+7\alpha+7\beta)^{20}=m^{10}\) then m is ___ .JEE Mains 2026 Easy
- Let \(\mathrm{f}: R \rightarrow R\) be function defined as \(f ( x )=\left\{\begin{array}{cc}3\left(1-\frac{| x |}{2}\right) & \text { if }| x | \leq 2 \text { } \\ 0 & \text { if }| x |>2 \text { }\end{array}\right.\) Let \(g: R \rightarrow R\) be given by \(g(x)=f(x+2)-f(x-2)\). If \(n\) and \(m\) denote the number of points in \(R\) where \(\mathrm{g}\) is not continuous and not differentiable, respectively, then \(\mathrm{n}+\mathrm{m}\) is equal to \(....\)JEE Mains 2021 Hard
- Let \(\overrightarrow{\mathrm{a}}=2 \hat{i}-\hat{j}+3 \hat{k}, \overrightarrow{\mathrm{~b}}=3 \hat{i}-5 \hat{j}+\hat{k}\) and \(\overrightarrow{\mathrm{c}}\) be a vector such that \(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}\) and \((\vec{a}+\vec{c}) \cdot(\vec{b}+\vec{c})=168\). Then the maximum value of \(|\vec{c}|^2\) is :JEE Mains 2025 Medium