The length of the axis of the conic 9 x²+4 y²-6 x+4 y+1=0 are

Correct Answer is : 1,231,2/31,32

Correct Answer is : 1,231,2/31,32

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Manipal Entrance Test 2019 Math Paper

Question : 13 of 70

Marks: +1, -0

9 x^{2}+4 y^{2}-6 x+4 y+1=0

are

Solution:

Given that equation of conic is

9 x^{2}+4 y^{2}-6 x+4 y+1=0

\Rightarrow 9 x^{2}-6 x+4 y^{2}+4 y+1=0

\Rightarrow 9 \left(x-\frac{1}{3}\right)^{2}+4 \left(y+\frac{1}{2}\right)^{2}=1

or,

\frac{\left(x-\frac{1}{3}\right)^{2}}{\frac{1}{9}}+\frac{\left(y+\frac{1}{2}\right)^{2}}{\frac{1}{4}}=1

Comparing it with standard equation of ellipse

\frac{X^{2}}{a^{2}}+\frac{Y^{2}}{b^{2}}=1,

we get

a=\frac{1}{3}, b=\frac{1}{2}

Length of major axis

=2 b=2 \times \frac{1}{2}=1

and length of minor axis

=2 a=2 \times \frac{1}{3}= \frac{2}{3}

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