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Ellipse

Section: Mathematics

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Question : 48 of 117

Marks: +1, -0

Let the ellipse

E: x^2+9 y^2=9

intersect the positive

x

y

-axes at the points

A

B

respectively. Let the major axis of

E

be a diameter of the circle

C

. Let the line passing through

A

B

meet the circle

C

P

. If the area of the triangle with vertices

A, P

O

\;\frac{m}{n}

m

n

are coprime, then

m-n

[10-Apr-2023 shift 1]

16
15
18
17

Solution:

Equation of line

AB

or

AP

is

\;\;\frac{x}{3}+\;\frac{y}{1}=1

\; x+3 y=3

\; x=(3-3 y)

Intersection point of line AP & circle is

P (x_0, y_0)

\; x^2+y^2=9 \Rightarrow (3-3y)^2+y^2=9

\; \Rightarrow 3^2 (1+y^2-2y)+y^2=9

\; \Rightarrow 5 y^2-9 y=0 \Rightarrow y(5y-9)=0

\; \Rightarrow y=\frac{9}{5}

Hence

x=3(1-y)=3 \left(1-\;\frac{9}{5}\right)=3 \left(\;\frac{-4}{5}\right)

\; x=\;\frac{-12}{5}

\; P(x_0, y_0)=\left(\;\frac{-12}{5},\;\frac{9}{5}\right)

Area of

\triangle AOP

is

=\;\frac{1}{2} \times OA \times

height

\; \text{ Height } \;=\frac{9}{5},\;\; OA=3

=\;\frac{1}{2} \times 3 \times \;\frac{9}{5}=\;\frac{27}{10}=\;\frac{m}{n}

Compare both side

m=27, n=10 \Rightarrow m-n=17

Therefore, correct answer is option-D

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