Question

The locus of mid-point of the segment intercepted between coordinate axes by any tangent to the ellipse $x^2/16+y^2=1$ is

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Answer 1

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Answer 2

Answer:

Let mid-point of part PQ which is in between the axis is R(x

1

,y

1

), then coordinates of P and Q will be (2x

1

,0) and (0,2y

1

), respectively.

∴ Equation of line PQ is

2x

1

x

+

2y

1

y

=1

⇒y=−(

x

1

y

1

)x+2y

1

If this line touches the ellipse

x

2

a

2

+

y

2

b

2

=1

then it will satisfy the condition,

c

2

=a

2

m

2

+b

2

So, (2y

1

)

2

=a

2

(

x

1

−y

1

)

2

+b

2

⇒4y

1

2

={

x

1

2

a

2

y

1

2

}+b

2

⇒4=(

x

1

2

a

2

)+(

y

1

2

b

2

)⇒(

x

1

2

a

2

)+(

y

1

2

b

2

)=4

∴ Required locus of (x

1

,y

1

) is

x

2

a

2

+

y

2

b

2

=4