Question

No tangent can be drawn from the point `(5/2,1)` to the circumcircle of the triangle with vertices `(1,sqrt(3)),(1,-sqrt(3)),(3,-sqrt(3))` .

Answer 1

Since, `(1,sqrt(3))`, `(1,-sqrt(3))` and `(3,sqrt(3))` form a right angled triangle at `(1,sqrt(3))`
`:.` Equation of circumcircle taking `(3,sqrt(3))` and `(1,-sqrt(3)` as end points of diameter.
`:.(x-3)(x-1)+(y-sqrt(3))(y+sqrt(3))=0`
`impliesx^(2)-4x+3+y^(2)-3=0`
`impliesx^(2)+y^(2)-4x=0`
At point `((5)/(2),1)`, `S_(1)=(25)/(4)+1-10 lt 0`
`:.` Point `(5//2,1)` lies inside the circle.
Hence, no tangent can be drawn.
Hence, given statment is true.