A line meets the co-ordinate axes in A, & | vedclass

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Question

Let the line $\frac{x}{a}+\frac{y}{b}=1$ meets the co-ordinate axes in $A$ and $B$

$A=(a, 0) ; B=(0, b)$

Let $C$ be the centre of the circle cir

Vedclass · Accepted Answer

$d_1 + d_2$

Vedclass · Answer

Let the line $\frac{x}{a}+\frac{y}{b}=1$ meets the co-ordinate axes in $A$ and $B$

$A=(a, 0) ; B=(0, b)$

Let $C$ be the centre of the circle circumscribing $	riangle O A B$. $C=\left(\frac{a}{2}, \frac{b}{2}ight)$ (Since circumcentre is the midpoint of hypotenuse in a right angled triangle) Radius $r=\frac{\sqrt{a^{2}+b^{2}}}{2}$

OC is perpendicular to the tangent at O. since, slope of $O C$ is $\frac{b}{a},$ slope of tangent at $O$ is $-\frac{a}{b}$ Equation of tangent is $a x+b y=0$

Let $d_{1}$ and $d_{2}$ be distances from $A$ and $B$ to the tangent respectively. $ightarrow d_{1}=\frac{a^{2}}{\sqrt{a^{2}+b^{2}}}$ and $d_{2}=\frac{b^{2}}{\sqrt{a^{2}+b^{2}}}$

$ightarrow d_{1}+d_{2}=\sqrt{a^{2}+b^{2}}=2 r$

Hence, diameter of the circle is $d_{1}+d_{2}$

A line meets the co-ordinate axes in $A\, \& \,B. \,A$ circle is circumscribed about the triangle $OAB.$ If $d_1\, \& \,d_2$ are the distances of the tangent to the circle at the origin $O$ from the points $A$ and $B$ respectively, the diameter of the circle is :

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