The correct option is A (2, 3)
Any point on the ellipse is given by (√8cosθ,√18sinθ) Now 2x8+218ydydx=0⇒dydx=−9x4y
⇒dydx|(√8cosθ,√18sinθ)=−9√8cosθ4√18sinθ=−√92cotθ.
Hence the equation of the tangent at (√18cosθ,√18sinθ) is
y - √18sinθ=−√92cotθ(x−√8cosθ)
Therefore, the tangent cuts the coordinates axes at the points (0,√18sinθ) and (√18cosθ,0)
Thus the area of the triangle formed by this tangent and the coordinate axes is A = 12√18√8 . 1cosθsinθ=6cosθsinθ=12cosec2θ.
But cosec2θ is smallest when θ=π4. Therefore A is smallest when θ=π4.
Hence the required point is (√8.1√2,√18.1√2) = (2, 3)