Let the ellipse E:x^2 + 9y^2 = 9 intersect the positive x and y-axes at the points A and B…

Question

Let the ellipse $E:{x^2} + 9{y^2} = 9$ intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. I...

Accepted Answer

The given equation of the ellipse is

$\begin{gathered} {x}^{2}+9{y}^{2}=9..........(i) \ \Rightarrow \frac{{x}^{2}}{9}+\frac{{y}^{2}}{1}=1 \end{gathered}$

Now, equation of line $AB$ is

$x+3y=3$...........(ii)

Now, point of intersection of line $x+3y=3$ and circle

$\begin{gathered} {x}^{2}+{y}^{2}=9 \ 	herefore (3-3y{)}^{2}+{y}^{2}=9 \ \Rightarrow 9-18y+9{y}^{2}+{y}^{2}=9 \ \Rightarrow -18y+10{y}^{2}=0 \ \Rightarrow 18y=10{y}^{2} \ \Rightarrow y=0,\frac{9}{5} \end{gathered}$

Now, from figure, we clearly see that vertices $A,P$ and $O$ makes a $	riangleAPO$,

whose area is $\frac{1}{2}	imes$ Base $	imes$ Height

$\begin{gathered} =\frac{1}{2}	imes 3	imes \frac{9}{5}=\frac{27}{10}=\frac{m}{n}[Given] \ 	herefore m=27,n=10 \ \Rightarrow m-n=27-10=17 \end{gathered}$

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