The tangent at a point P on the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1, which in not an extremely of major axismeets a directrix at T. Statement-1: The circle on PT as diameter passes through the focus of the ellipse corresponding to the directrix on which T lies. Statement-2: Pt substends is a right angle at the focus of the ellipse corresponding to the directrix on which T lies.

The tangent at a point P on the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)

1.	The tangent at a point P on the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1, which in not an extremely of major axismeets a directrix at T. Statement-1: The circle on PT as diameter passes through the focus of the ellipse corresponding to the directrix on which T lies. Statement-2: Pt substends is a right angle at the focus of the ellipse corresponding to the directrix on which T lies.
Answer» Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1 Statement-1 is True, Statement-2 is True, Statement -2 is not a correct explanation for Statement-1 Statement-1 is True, Statement-2 is False. Statement-1 is False, Statement-2 is True Solution :The equation of tangents at a POINT P `(x_(1),y_(1))` on the ellipse`(x^(2))/(a^(2))+(y^(2))/(b^(2))=1 " is " ("xx"_(1))/(a^(2))+(yy_(1))/(b^(2))=1`. This cuts the directrix `x=(a)/(e) " at " T((a)/(e),((AE-x_(1))b^(2))/(aey_(1)))` The coordinates of the focus S are (ae, 0). `therefore m_(1)= "Slope of Sp"=(y_(1))/(x_(1)-ae)` and, `m_(2)="Slope of ST " =((ae-x_(1))b^(2))/(aey_(1)(a//e-ae))=(ae-x_(1))/(y_(1))` Clearly, ` m_(1)m_(2)=-1`. So, PT substends a right ANGLE at the focus, Consequently,circle described on PT as a DIAMTER passes through the focus of the ellipse. HENCE, both the statement ar true and Statement -2 is a correct explanation for Statement-1.

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