Question

In a triangle ABC, C = 90', then the equation
whose roots are tanA, tanB is
1) ab x² +c²x + ab = 0 2) ab x² + c²x - ab = 0
3) ab x² - c²x - ab = 0 4) ab x²-c²x + ab = 0​

Answer 1

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Given:

In a Triangle ABC,C = 90

To find:

The equation whose roots are TanA, Tan B is

Solution:

From given, we have,

The sum of angles of a triangle,

A + B + C = 180

Given, C = 90

A + B + 90 = 180

A + B = 90

tan (A + B) = tan A + tan B / 1 - tan A × tan B

as A + B = 90, we get,

tan 90 = tan A + tan B / 1 - tan A × tan B

1 / 0 = tan A + tan B / 1 - tan A × tan B

1 - tan A × tan B = 0

∴ tan A × tan B = 1

the quadratic equation,

x² + sum of roots x + product of roots = 0

x² - (tan A + tan B)x + tan A × tan B = 0

tan A = a/b and tan B = b/a

tan A + tan B = a/b + b/a = (a²+b²)/ab

(a² + b² = c²) Pythagoras theorem.

∴ tan A + tan B= c²/ab

x² - (tan A + tan B)x + tan A × tan B = 0

x² - (c²/ab)x + 1 = 0

abx² - c²x + ab = 0

Therefore, the required equation is, abx² - c²x + ab = 0.

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