Tangents are drawn through the point (4,√3) to the ellipse x216+y29=1. The points at which these tangents touch the ellipse are
A
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B
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C
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D
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Solution
The correct option is A Let P(x1,y1) be the point of contact. The equation of the tangent at P is xx116+yy19=1 If this tangent passes through (4,√3), then x14+y13√3=1⇒y13√3=1−x14⇒y1=12√3−3√3x14 P lies on the ellipse ⇒x2116+y219=1⇒x2116+(12√3−3√3x1)2144=1⇒x21+(4√3−√3x1)2=16 ⇒x21+48+3x21−24x1=16⇒4x21−24x1+32=0⇒x21−6x1+8=0⇒x1=2or4 x1=2⇒y1=3√32,x1=4⇒y1=0 ∴ The required points are (2,3√32),(4,0)