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$If tangents are drawn to the ellipse x^{2} + 2 y^{2} = 2, then the locus of the midpoint of$
$the intercept made by the tangents between the coordinate axes is$

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a

12x2+14y2=1

b

14x2+12y2=1

c

x22+y24=1

d

x24+y22=1

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detailed solution

Correct option is A

Tangent to ellipsex22+y21=1 at P(2cos⁡θ,sin⁡θ) is given byxcos⁡θ2+ysin⁡θ=1It meets axes at A and B . Let Q(h, k) be midpoint of AB. Using midpoint formula, we haveA≡(2sec⁡θ,0) and B≡(0,cosec⁡θ) Hence ,2h=2sec⁡θ and 2k=cosec⁡θor 12h2+12k2=1or 12h2+14k2=1or12x2+14y2=1

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