First slide

Tangents and normals

Question

Segment of the tangent to the curve $x y = c^{2}$ at the points $(x', y')$ which is contained between the co-ordinate axes, is bisected at the point:

Moderate

A

$(- x', - y')$
B

$(y', x')$
C

$(x' / 2, y' / 2)$
D

None

Solution

${[\frac{d y}{d x}]}_{(x', y')} = - \frac{c^{2}}{x'^{2}} = - \frac{y'}{x'}$
$∴$ Equation of tangent at $(x', y')$ is $y - y' = (- y' / x') (x - x')$

Which meets the co-ordinates axes at $A$ and $B$ (say).

Then $A = (2 x', 0), B = (0, 2 y')$ .

Mid-point of $A B = (x', y')$ .

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