AB is a variable chord of the ellipse frac{x^ | vedclass

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(A) Let the equation of the chord $AB$ be $lx + my = 1$. Homogenizing the equation of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with the hel

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$\frac{1}{a^2} + \frac{1}{b^2}$

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(A) Let the equation of the chord $AB$ be $lx + my = 1$. Homogenizing the equation of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with the help of the chord equation $lx + my = 1$, we get: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = (lx + my)^2$ $\frac{x^2}{a^2} + \frac{y^2}{b^2} = l^2x^2 + m^2y^2 + 2lmxy$ $x^2(\frac{1}{a^2} - l^2) + y^2(\frac{1}{b^2} - m^2) - 2lmxy = 0$ Since $AB$ subtends a right angle at the origin, the sum of the coefficients of $x^2$ and $y^2$ must be zero. $(\frac{1}{a^2} - l^2) + (\frac{1}{b^2} - m^2) = 0$ $l^2 + m^2 = \frac{1}{a^2} + \frac{1}{b^2}$ Now, the distance of the chord $lx + my = 1$ from the origin is $p = \frac{1}{\sqrt{l^2 + m^2}}$. If $A$ and $B$ are points on the ellipse, the coordinates can be represented such that $OA^2$ and $OB^2$ relate to the chord properties. For a chord subtending a right angle at the origin, the expression $\frac{1}{OA^2} + \frac{1}{OB^2}$ is constant and equal to $\frac{1}{a^2} + \frac{1}{b^2}$.

$AB$ is a variable chord of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. If $AB$ subtends a right angle at the origin $O$, then $\frac{1}{OA^2} + \frac{1}{OB^2}$ equals to

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