Answer
Verified
396.9k+ views
Hint: From the given two equations we first try to find out the center of the circle and its radius. And then we have to find out if the parabola and the circle are touching each other or not. Thus we can find the result by observing the solution.
Complete step by step answer:
Here in this problem, we are given two equations, \[{x^2} + {y^2} + 4ax = 0\] and \[{y^2} = 4ax\]. Where one of them is a circle and the other is a parabola. We need to find the equation of common tangents of these two.
So, the equation of the circle is,
\[{x^2} + {y^2} + 4ax = 0\]
Adding \[4{a^2}\] to both sides we get,
\[ \Rightarrow {x^2} + 4ax + 4{a^2} + {y^2} = 4{a^2}\]
On applying \[{{\text{a}}^{\text{2}}}{\text{ + 2ab + }}{{\text{b}}^{\text{2}}}{\text{ = (a + b}}{{\text{)}}^{\text{2}}}\], we get,
\[ \Rightarrow {(x + 2a)^2} + {y^2} = 4{a^2}\]……(i)
So, we get the circle with center at \[( - 2a,0)\]and with radius \[2a\] units.
And we have the 2nd equation as, \[{y^2} = 4ax\] which is passing through the origin.
Now, if we try to plot them altogether, we get,
Now, from the figure, it is clearly visible that, the circle and the parabola are touching each other along the y-axis and it is the common tangent of these two given equations.
So, now, the equation of the common tangent is, \[x = 0\]
Note: In this given problem we focused on finding where the different figures are meeting each other and if they are at all touching each other or not. In these types of problems, if the figures are touching each other, our job becomes easier as the common tangent is always passing through their point of intersection.
Complete step by step answer:
Here in this problem, we are given two equations, \[{x^2} + {y^2} + 4ax = 0\] and \[{y^2} = 4ax\]. Where one of them is a circle and the other is a parabola. We need to find the equation of common tangents of these two.
So, the equation of the circle is,
\[{x^2} + {y^2} + 4ax = 0\]
Adding \[4{a^2}\] to both sides we get,
\[ \Rightarrow {x^2} + 4ax + 4{a^2} + {y^2} = 4{a^2}\]
On applying \[{{\text{a}}^{\text{2}}}{\text{ + 2ab + }}{{\text{b}}^{\text{2}}}{\text{ = (a + b}}{{\text{)}}^{\text{2}}}\], we get,
\[ \Rightarrow {(x + 2a)^2} + {y^2} = 4{a^2}\]……(i)
So, we get the circle with center at \[( - 2a,0)\]and with radius \[2a\] units.
And we have the 2nd equation as, \[{y^2} = 4ax\] which is passing through the origin.
Now, if we try to plot them altogether, we get,
Now, from the figure, it is clearly visible that, the circle and the parabola are touching each other along the y-axis and it is the common tangent of these two given equations.
So, now, the equation of the common tangent is, \[x = 0\]
Note: In this given problem we focused on finding where the different figures are meeting each other and if they are at all touching each other or not. In these types of problems, if the figures are touching each other, our job becomes easier as the common tangent is always passing through their point of intersection.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What organs are located on the left side of your body class 11 biology CBSE
Write an application to the principal requesting five class 10 english CBSE
What is the type of food and mode of feeding of the class 11 biology CBSE
Name 10 Living and Non living things class 9 biology CBSE