MHT CET · Maths · Application of Derivatives
All the points on the curve \(y^{2}=4 a|x+a \sin (x / a)|\), where the tangent is parallel to the axis of \(x\) are lies on
- A circle
- B parabola
- C straight line
- D None of these
Answer & Solution
Correct Answer
(B) parabola
Step-by-step Solution
Detailed explanation
\(y^{2}=4 a\left[x+a \sin \left(\frac{x}{a}\right)\right]\)...(i)
\(\therefore 2 y \frac{d y}{d x}=4 a\left[1+\cos \left(\frac{x}{a}\right)\right]\)...(ii)
If tangent is parallel to \(x\) -axis, then \(\frac{d y}{d x}=0\)
So, from Eq. (i), we get \(\cos \left(\frac{x}{a}\right)=-1\)
\(\therefore\) \(
\sin \left(\frac{x}{a}\right)=0
\)
On putting this value in Eq. (i), we get \(y^{2}=4 a(x+0) \Rightarrow y^{2}=4 a x\)
So, all the points on the curve
\(
y^{2}=4 a\left(x+a \sin \frac{x}{a}\right)
\)
where the tangent is parallel to the \(x\) -axis are lies on parabola.
\(\therefore 2 y \frac{d y}{d x}=4 a\left[1+\cos \left(\frac{x}{a}\right)\right]\)...(ii)
If tangent is parallel to \(x\) -axis, then \(\frac{d y}{d x}=0\)
So, from Eq. (i), we get \(\cos \left(\frac{x}{a}\right)=-1\)
\(\therefore\) \(
\sin \left(\frac{x}{a}\right)=0
\)
On putting this value in Eq. (i), we get \(y^{2}=4 a(x+0) \Rightarrow y^{2}=4 a x\)
So, all the points on the curve
\(
y^{2}=4 a\left(x+a \sin \frac{x}{a}\right)
\)
where the tangent is parallel to the \(x\) -axis are lies on parabola.
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 line \(x-2 y=\mathrm{m}(\mathrm{m} \in \mathrm{Z})\) intersects the circle \(x^2+y^2=2 x+4 y\) at two distinct points, then the number of possible values of \(\mathrm{m}\) areMHT CET 2023 Medium
- Let the vectors \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) be such that \(|\overline{\mathrm{a}}|=2,|\dot{\bar{b}}|=4\) and \(|\overline{\mathrm{c}}|=4\). If the projection of \(\overline{\mathrm{b}}\) on \(\overline{\mathrm{a}}\) is equal to the projection of \(\overline{\mathrm{c}}\) on \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) is perpendicular to \(\overline{\mathrm{c}}\), then the value of \(|\overline{\mathrm{a}}+\overline{\mathrm{b}}-\overline{\bar{c}}|\) is equal toMHT CET 2024 Easy
- It is required to seat 5 men and 4 women in a row so that the men occupy odd places. Then the number of arrangements that are possible isMHT CET 2022 Easy
- The logical statement \((p \wedge \sim q) \vee q \vee(\sim p \wedge q)\) is equivalent toMHT CET 2022 Easy
- \(\int \frac{\mathrm{d} x}{\cot ^2 x-1}=\frac{1}{\mathrm{~A}} \log |\sec 2 x+\tan 2 x|-\frac{x}{\mathrm{~B}}+\mathrm{c}\), (where \(\mathrm{c}\) is constant of integration), then \(\mathrm{A}+\mathrm{B}=\)MHT CET 2023 Hard
- If the slope of one of the lines given by \(a x^2+2 h x y+b y^2=0\) is two times the other, thenMHT CET 2022 Easy
More PYQs from MHT CET
- If slope of one of the lines \(a x^2+2 h x y+b y^2=0\) is twice that of the other, then \(\mathrm{h}^2: \mathrm{ab}\) isMHT CET 2021 Medium
- What is the product obtained in Reimer - Tiemann reaction?MHT CET 2022 Medium
- The equation of the plane passing through the points \((1,2,3)\), \((-1,4,2)\) and \((3,1,1)\) isMHT CET 2022 Easy
- When photons of energy fall on metal plate of work function ‘ ’, photoelectrons of maximum kinetic energy, ‘ ’ are ejected. If the frequency of the radiation is doubled, the maximum kinetic energy of the ejected photoelectrons will beMHT CET 2019 Medium
- Figure shows two semicircular loops of radii \(R_1\) and \(R_2\) carrying current \(I\). The magnetic field at the common centre ' \(\mathrm{O}\) ' is
MHT CET 2023 Hard - The primary and secondary voltage of an ideal step-down transformer is \(200 \mathrm{~V}\) and \(25 \mathrm{~V}\) respectively. The secondary is connected to a device, which draws a current of \(2 \mathrm{~A}\). The current in the primary isMHT CET 2022 Easy