Question

An ellipse with major and minor axis length as $2a$ and $2b$ units touches coordinate axis in first quadrant. If foci are $(x_1,y_1)$ and $(x_2,y_2),$ then the value of $x_1{x}_2+y_1{y}_2$ is

• A. $2a^2$
• B. $2b^2$
• C. $a^2{b}^2$
• D. $a^2+b^2$

Answer 1

The correct option is B $2b^2$

We know that product of length of perpendicular from foci upon any tangents to ellipse is $b^2.$

Hence from the diagram, $x_1$ and $x_2$ are the length of perpendiculars from foci upon tangent $y-$ axis of the given ellipse, thus $x_1{x}_2=b^2$
Similarly $y_1{y}_2=b^2$
$\therefore x_1{x}_2+y_1{y}_2=2b^2$