Ellipse 2 Question 13

13. Tangents are drawn to the ellipse x2+2y2=2, then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is

(2004, 1M)

(a) 12x2+14y2=1

(b) 14x2+12y2=1

(c) x22+y24=1

(d) x24+y22=1

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Solution:

  1. Let the point P(2cosθ,sinθ) on x22+y21=1.

446 Ellipse

Equation of tangent is, x22cosθ+ysinθ=1

whose intercept on coordinate axes are

A(2secθ,0) and B(0,cosecθ)

Mid-point of its intercept between axes

22secθ,12cosecθ=(h,k)cosθ=12h and sinθ=12k

Thus, focus of mid-point M is

(cos2θ+sin2θ)=12h2+14k212x2+14y2=1, is required locus. 



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