Question

If a tangent on ellipse at $A(1,1)$ intersects its directrix at $B(7,–7)$ and $S$ be the corresponding focus of ellipse and $C(\alpha ,\beta )$ is the circumcentre of $△SAB,$ then

• A. $\alpha +\beta =1$
• B. $\alpha -\beta =7$
• C. $S{C}^2=20.5$
• D. $S{C}^2=25$

Answer 1

Answer: (A), (B), (D)

$S{C}^2=25$

$△SAB$ is right angled at $S.$
Circumcentre $C$ is mid point of hypotenuse
$C\equiv (\frac{1+7}{2},\frac{1-7}{2})=(4,-3)$
$\therefore \alpha =4$ and $\beta =-3$
$\therefore \alpha +\beta =1$ and $\alpha -\beta =7$

$SC$ is radius and $AB$ is diameter.
$\therefore SC=\frac{1}{2}\sqrt{36+64}=\frac{1}{2}\sqrt{100}=5$