Question

Find the equation of tangent to the ellipse \[4{{x}^{2}}+9{{y}^{2}}=72\] which is perpendicular to the line \[3x-2y=5\]

Answer 1

Hint: First we have to convert the given ellipse equation to the general form and obtain the values of\[{{a}^{2}}\]and \[{{b}^{2}}\]. We have know that if two lines are perpendicular then the product of their slopes is equal to \[-1\]

Complete step-by-step answer:

Given the equation of ellipse is \[4{{x}^{2}}+9{{y}^{2}}=72\] and we have to find the tangent which is perpendicular to the line \[3x-2y=5\]
Firstly convert the given ellipse equation to general form that is \[\dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\]
So by converting the given equation to the general form we will get equation as follows
\[\dfrac{{{x}^{2}}}{18}+\dfrac{{{y}^{2}}}{8}=1\]
So by comparing the obtained equation with general form we will get the following values,
\[{{a}^{2}}=18\]
\[{{b}^{2}}=8\]
We know that the equation of tangent to the ellipse in slope form is
\[y=mx+\sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}\]
But in the question given that tangent is perpendicular to the line \[3x-2y=5\]
So slope of the line \[3x-2y=5\]is \[\dfrac{3}{2}\]
We know that when two lines are perpendicular to each other then \[\mathop{m}_{1}\times \mathop{m}_{2}=-1\]
\[\dfrac{3}{2}\times \mathop{m}_{2}=-1\]
\[\mathop{m}_{2}=\dfrac{-2}{3}\]
So the slope of the tangent to the given ellipse equation is \[\dfrac{-2}{3}\]
Now substitute \[m=\dfrac{-2}{3}\] in the equation of tangent to the ellipse in slope form we will get
\[y=mx+\sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}\]
Now substitute the obtained values in the given expression we will get,
\[y=\dfrac{-2}{3}x+\sqrt{18\left( \dfrac{4}{9} \right)+8}\]
\[y=\dfrac{-2}{3}x+\sqrt{16}\]
One equation of tangent to the ellipse is
\[y=\dfrac{-2}{3}x+4\]
\[2x+3y-12=0\]
Other equation of tangent to the ellipse is
\[y=\dfrac{-2}{3}x-4\]
\[2x+3y+12=0\]
Hence we get the required equation of tangents to the given ellipse
Note: We have to note that general equation of tangents to any given ellipse is of the form \[y=mx+\sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}\]and in the general form of the ellipse\[\dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\] if \[{{a}^{2}}\] is greater than \[{{b}^{2}}\]then a is called as major axis and b is called as minor axis in ellipse