If-tangent-line-of-equation-y-x-3-x-at-x-a-crossed-line-y-x-at-b-b-Find-b-in-terms-of-a-

Question Number 19812 by Joel577 last updated on 16/Aug/17

If tangent line of equation y = \frac{x}{3 - x} at

x = a crossed line y = x at (b, b)

Find b in terms of a

Answered by ajfour last updated on 16/Aug/17

y = - 1 - \frac{3}{x - 3} \Rightarrow \frac{dy}{dx} = \frac{3}{{(x - 3)}^{2}}

\frac{dy}{dx} ∣_{x = a} = \frac{3}{{(a - 3)}^{2}}

Equation of tangent at x = a is

y - y_{1} = (\frac{dy}{dx}) ∣_{x = a} (x - a)

y_{1} = \frac{a}{3 - a}, so

y - (\frac{a}{3 - a}) = \frac{3}{{(a - 3)}^{2}} (x - a)

this tangent crosses y = x at (b, b)

so (b, b) must satisfy tangent eqn .

b - \frac{a}{(3 - a)} = \frac{3 (b - a)}{{(a - 3)}^{2}}

\frac{3 b}{{(a - 3)}^{2}} - b = \frac{3 a}{{(a - 3)}^{2}} + \frac{a}{(a - 3)}

at x = a = 3 there is no function and

no tangent, so

b [3 - {(a - 3)}^{2}] = a (3 + a - 3)

\Rightarrow b = \frac{a^{2}}{3 - {(a - 3)}^{2}} .

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