The length of the chord intercepted by the ci | vedclass

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Question

(B) The length of the chord is given by $L = 2\sqrt{r^2 - d^2}$,where $r$ is the radius and $d$ is the perpendicular distance from the center $(0, 0)$

Vedclass · Accepted Answer

$2\sqrt{\frac{r^2(a^2 + b^2) - a^2b^2}{a^2 + b^2}}$

Vedclass · Answer

(B) The length of the chord is given by $L = 2\sqrt{r^2 - d^2}$,where $r$ is the radius and $d$ is the perpendicular distance from the center $(0, 0)$ to the line $\frac{x}{a} + \frac{y}{b} - 1 = 0$.The distance $d = \frac{|\frac{0}{a} + \frac{0}{b} - 1|}{\sqrt{(\frac{1}{a})^2 + (\frac{1}{b})^2}} = \frac{1}{\sqrt{\frac{a^2 + b^2}{a^2b^2}}} = \frac{ab}{\sqrt{a^2 + b^2}}$.Substituting $d$ into the formula: $L = 2\sqrt{r^2 - (\frac{ab}{\sqrt{a^2 + b^2}})^2} = 2\sqrt{r^2 - \frac{a^2b^2}{a^2 + b^2}}$.Simplifying,we get $L = 2\sqrt{\frac{r^2(a^2 + b^2) - a^2b^2}{a^2 + b^2}}$.

The length of the chord intercepted by the circle $x^2 + y^2 = r^2$ on the line $\frac{x}{a} + \frac{y}{b} = 1$ is:

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