Find the equations of the tangent and normal to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1`. at the point `(x_0,y_0)`

Explanation

Equation of hyperbola
` x^(2)/a^(2) - y^(2)/b^(2) = 1` …(1)
` rArr (2x)/a^(2) - (2y)/b^(2) (dy)/(dx) = 0 rArr (dy)/(dx) = (xb^(20)/(ya^(2))`
Point `(x_(0), y_(0))` lines on curve (1)
` :. (x_(0)^(2))/a^(2) - (y_(0)^(2))/b^(2) = 1`
at point ` (x_(0), y_(0))`
Slope of tangent = `(b^(2)x_(0))/(a^(2)y_(0))`
and equation of tangent
` y - y_(0) = (b^(2)x_(0))/(a^(2)y_(0))(x-x_(0))`
` rArr (y y_(0))/b^(2) - (y_(0)^(2))/b^(2) = (xx_(0))/a^(2)`
` rArr (x x_(0))/a^(2)- (y y_(0))/b^(2) = (x_(0)^(2))/a^(2) - (y_(0)^(2))/b^(2)`
` rArr (x x_(0))/a^(2) - (y y_(0))/b^(2) = 1` from equation (2)
Slope of normal = `-(a^(2)y_(0))/(b^(2)x_(0))`
and equation of normal
` y-y_(0)= - (a^(2)y_(0))/(b^(2)x_(0)) (x-x_(0))`
` rArr (y-y_(0))/(a^(2)y_(0))=-(x-x_(0))/(b^(2)x_(0))` ,brgt ` rArr (x-x_(0))/(b^(2)x_(0))+(y-y_(0))/(a^(2)y_(0)) = 0`