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The line ax+by=7ax + by = 7 is a tangent to the curve y=x−7(x−2)(x−3)y = \frac{x-7}{(x-2)(x-3)} at the point where it cuts the x-axis

A. The y-intercept of the line is −0.7-0.7

B. b=−7b = -7

C. a=1a = 1

D. the line passes through the point (−13,−1)(-13, -1)

E. b=−20b = -20

Choose the correct answer from the options given below:

Solution

✅ Correct Option: 2

Curve cuts x-axis at (7,0)(7, 0).

Slope at (7,0)(7,0): using quotient rule, dydx=1(7−2)(7−3)=120\frac{dy}{dx} = \frac{1}{(7-2)(7-3)} = \frac{1}{20}.

Tangent: y=120(x−7)y = \frac{1}{20}(x - 7), i.e. x−20y=7x - 20y = 7. So a=1a = 1, b=−20b = -20.

Check D: −13−20(−1)=7-13 - 20(-1) = 7, holds. C, D, E correct.

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