If a line passing through the point P(1, 1) m | vedclass

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(C) The equation of the line passing through $P(1, 1)$ with angle $	heta$ is $\frac{x - 1}{\cos	heta} = \frac{y - 1}{\sin	heta} = r$.The coordinate

Vedclass · Accepted Answer

$1 + \sqrt{3}$

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(C) The equation of the line passing through $P(1, 1)$ with angle $	heta$ is $\frac{x - 1}{\cos	heta} = \frac{y - 1}{\sin	heta} = r$.The coordinates of $A$ (where $y=0$) are $(1 - \cot	heta, 0)$ and $B$ (where $x=0$) are $(0, 1 - 	an	heta)$.Given $AB^2 = OP^2$,we have $(1 - \cot	heta)^2 + (1 - 	an	heta)^2 = 1^2 + 1^2 = 2$.Expanding this: $(1 - 2\cot	heta + \cot^2	heta) + (1 - 2	an	heta + 	an^2	heta) = 2$.$2 - 2(\cot	heta + 	an	heta) + (\cot^2	heta + 	an^2	heta) = 2$.Let $u = 	an	heta + \cot	heta$. Since $\cot^2	heta + 	an^2	heta = u^2 - 2$,we get:$u^2 - 2 - 2u = 0 \Rightarrow u^2 - 2u - 2 = 0$.Solving for $u$: $u = \frac{2 \pm \sqrt{4 - 4(1)(-2)}}{2} = \frac{2 \pm \sqrt{12}}{2} = 1 \pm \sqrt{3}$.Since $|	an	heta + \cot	heta| \ge 2$,the value $1 - \sqrt{3}$ is rejected as it is approximately $-0.732$.Thus,$	an	heta + \cot	heta = 1 + \sqrt{3}$.

If a line passing through the point $P(1, 1)$ makes an angle $\theta$ with the positive direction of the $x$-axis and intersects the coordinate axes at $A$ and $B$ such that $AB = OP$,where $O$ is the origin,then what is the value of $\tan\theta + \cot\theta$?

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