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The tagents at any point P of the circle x^(2)+y^(2)=16 meets the tangents at a fixed point A at T. Point is joined to B,the other end of the diameter, through, A. The locus of the intersection of AO and BT is a conic whosee eccentricity is |
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Answer» `1//2` The equation of AP is `y=(sin theta)/(cos theta-1)(x-4)""(2)` From (1), the coordinates of the point T are given by `(4,(4(1-costheta))/(sin theta))` The equation of BT is `y=(1-cos theta)/(2 sin theta)(x+4)""(3)` Let (h,k) be the point of intersection of the lines (2), and (3). Then we have `k^(2)=-(1)/(2)(h^(2)-16)` or `(h^(2))/(16)+(y^(2))/(8)=1` Therefore, the locus of (h,k) is `(x^(2))/(16)+(y^(2))/(8)=1` Which is an ellipse with eccehntrically `e=1//sqrt(2)` Sumof FOCAL distance of any points is 2a=8 Considering the circle `x^(2)+y^(2)=a^(2)`, we find the that the ECCENTRICITY of the ellipse is `1sqrt(2)` which is contant and does not CHANGE by CHANGING the radius of the circle |
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