MHT CET · Maths · Parabola
The equation of the common tangent touching the circle \((x-3)^{2}+y^{2}=9\) and the parabola \(y^{2}=4 x\) above the \(x\) -axis is
- A \(\sqrt{2} y=3 x+1\)
- B \(\sqrt{3} y=-(x+3)\)
- C \(\sqrt{3} y=x+3\)
- D \(\sqrt{3} y=-(3 x+1)\)
Answer & Solution
Correct Answer
(C) \(\sqrt{3} y=x+3\)
Step-by-step Solution
Detailed explanation
Let the common tangent to the parabola
\(
\begin{array}{l}
y^{2}=4 x \text { be } \\
y=m x+\frac{1}{m}
\end{array}
\)
It should be also touch the circle
\(
(x-3)^{2}+y^{2}=9
\)
whose centre is \((3,0)\) and radius \(=3\), then
\(
\begin{array}{cc}
& \frac{|3 m+1 / m|}{\sqrt{1+m^{2}}}=3 \\
\Rightarrow & 3 m^{2}=1 \\
\Rightarrow & \quad m=\pm \frac{1}{\sqrt{3}}
\end{array}
\)
But \(m>0\), then equation of common tangent is
\(
y=\frac{1}{\sqrt{3}} \cdot x+\sqrt{3}
\)
or \(\sqrt{3} \cdot y=x+3\)
\(
\begin{array}{l}
y^{2}=4 x \text { be } \\
y=m x+\frac{1}{m}
\end{array}
\)
It should be also touch the circle
\(
(x-3)^{2}+y^{2}=9
\)
whose centre is \((3,0)\) and radius \(=3\), then
\(
\begin{array}{cc}
& \frac{|3 m+1 / m|}{\sqrt{1+m^{2}}}=3 \\
\Rightarrow & 3 m^{2}=1 \\
\Rightarrow & \quad m=\pm \frac{1}{\sqrt{3}}
\end{array}
\)
But \(m>0\), then equation of common tangent is
\(
y=\frac{1}{\sqrt{3}} \cdot x+\sqrt{3}
\)
or \(\sqrt{3} \cdot y=x+3\)
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 \(\mathrm{f}(\mathrm{x})=[\mathrm{x}]\), for \(\mathrm{x} \in(-1,2)\), then \(\mathrm{f}\) is discontinuous at (where \([\mathrm{x}]\) represents floor function)MHT CET 2021 Easy
- In a class of 300 students, every student reads 5 news papers and every news paper is read by 60 students. Then the number of newspapers isMHT CET 2024 Easy
- If the function \(\mathrm{f}(x)=x(x+3) \mathrm{e}^{-\frac{x}{2}}\) satisfies all the conditions of Rolle's theorem in \([-3,0]\), then c isMHT CET 2025 Easy
- If \(f(x)=x, g(x)=\sin x\), then \(\int f(g(x)) d x\) is equal toMHT CET 2010 Easy
- Let a: and . If the truth values of p and q are true and that of r and s are false, then the truth values of a and b are respectively…MHT CET 2019 Easy
- In a triangle with one of the angles \(120^{\circ}\), the lengths of the sides form an A.P. If length of the greatest side is 7 m , then the area of the triangle isMHT CET 2025 Medium
More PYQs from MHT CET
- If \(\alpha\) and \(\beta\) are the complex cube roots of unity, then \(\alpha^3+\beta^3+\alpha^{-2} \times \beta^{-2}\) is equal toMHT CET 2022 Easy
- The parametric equations of the curve \(x^2+y^2+a x+b y=0\) areMHT CET 2023 Hard
- The ratio of the binding energy of a satellite at rest on earth's surface to the binding energy of a satellite of same mass revolving around the earth at a height above the earth's surface is = radius of the earth.MHT CET 2017 Medium
- DPD of a cell can be expressed as __________.MHT CET 2020 Hard
- Primary ectoderm during gastrulation develops from _________ .MHT CET 2024 Easy
- For simple cubic crystal edge length is expressed asMHT CET 2021 Easy