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The tangent at a point P on the ellipse ...

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 axis meets 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.

A

Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1

B

Statement-1 is True, Statement-2 is True, Statement -2 is not a correct explanation for Statement-1

C

Statement-1 is True, Statement-2 is False.

D

Statement-1 is False, Statement-2 is True

Text Solution

Verified by Experts

The correct Answer is:
A
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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