Questions & Answers

Question

Answers

Answer
Verified

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 2

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, \[x = 0\]