Question

if two distinct tangents can be drawn from a point (alfa,2) on different branches of hyperbola x square by 9 minus y square by 16 equal to 1 then Alpha belongs to​

Answer 1

Given : two distinct tangents can be drawn from a point (α,2) on different branches of hyperbola x²/9 - y²/16 = 1

To Find :  α belongs to​

Solution:

x²/9  - y²/16  = 1

=> x²/3² - y²/4² = 1

y = mx + c is tangent

c = ±√3²m² - 4²

=>  c =±√9m² - 16

y = mx  ± √9m² - 16

masses through α , 2

=> 2 = mα  ± √9m² - 16

=> 2 - mα = ± √9m² - 16

Squaring both sides

=> 4 + m²α² - 4mα = 9m² - 16

=> m²(α²- 9) -  4mα  + 20 = 0

Distinct real roots if

(-4α)² - 4(α²- 9)(20) > 0

=> 16α²  - 80α² + 720 > 0

=> 720 > 64α²

=>  45 >  4α²

=> α² < 45/4

=>     |α | < 3√5/2

=>     -3√5/2 <  α  < 3√5/2

Learn More

Find the equation of hyperbola x^2/36- y^2/9=1 at the point whose ...

https://brainly.in/question/14927699

the equation of tangent at point (x1,y1) having slope m1 is obtained by

https://brainly.in/question/24107768