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Let a & b be any two numbers satisfying ...

Let a & b be any two numbers satisfying `1/a^2 + 1/b^2 = 1/4`.Then, the foot of the perpendicular from the origin on variable line `x/a + y/b = 1` lies on

Text Solution

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We can draw the line `x/a+y/b = 1` on the coordinate axis.
Please refer to video to see the diagram.
Here, `AB` is the line. Let `D(h,k)` is the point that is foot of the perpendicular to line `AB`.
Then, Area of `Delta AOB = 1/2|OA*OB| = 1/2|ab|`
Also, Area of `Delta AOB = 1/2|OD*AB| = 1/2sqrt(a^2+b^2)sqrt(h^2+k^2)`
`:. 1/2|ab| = 1/2sqrt(a^2+b^2)sqrt(h^2+k^2)`

`=>a^2b^2 = (a^2+b^2)(h^2+k^2)`
`=>h^2+k^2 = (a^2b^2)/(a^2+b^2)`
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