A (a, 0) and B (-a, 0) are two fixed points o | vedclass

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(A) Let the coordinates of the fixed points be $A (a, 0)$ and $B (-a, 0)$,and let the moving point be $C (h, k)$.From the given figure,let $D$ be the

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$y\lambda = 2a$

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(A) Let the coordinates of the fixed points be $A (a, 0)$ and $B (-a, 0)$,and let the moving point be $C (h, k)$.From the given figure,let $D$ be the projection of $C$ on the $x$-axis,so $D = (h, 0)$.In $\Delta ADC$,$\cot A = \frac{AD}{CD} = \frac{a - h}{k}$.In $\Delta BDC$,$\cot B = \frac{BD}{CD} = \frac{h - (-a)}{k} = \frac{h + a}{k}$.Given $\cot A + \cot B = \lambda$,we have:$\frac{a - h}{k} + \frac{h + a}{k} = \lambda$$\frac{2a}{k} = \lambda$Replacing $(h, k)$ with $(x, y)$,we get $\frac{2a}{y} = \lambda$,which simplifies to $y\lambda = 2a$.

$A (a, 0)$ and $B (-a, 0)$ are two fixed points of $\Delta ABC$. If the vertex $C$ moves such that $\cot A + \cot B = \lambda$,where $\lambda$ is a constant,then what is the locus of point $C$?

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