Find the equation of tangents to circle `x^(2)+y^(2)-2x+4y-4=0`
drawn from point P(2,3).

Explanation

Equation of the circle is `x^(2)+y^(2)-2x+4y-4=0`.
Centre of the circle is C(1,-2) and radius is `sqrt((-1)^(2)+2^(2)-(-4))=3`.
Tangents are drawn to the circle from point P(2,3).
Equation of the line through point P having slope m is
`y-3=m(x-2)`
or `mx-y-2m+3=0`
If the line touches the circle , then the distance from the centre of the circle to this line will be radius of the circle.
`:. |(m(1)+2-2m+3)/(sqrt(m^(2)+1))|=3`
`implies (5-m)^(2)=9(m^(2)+1)`
`implies 8 m^(2)+10m-16=0`
`implies 4m^(2)+5m-8=0`
`implies m=(-5+- sqrt(153))/(8)`
Therefore, required equations of tangents are
`y=3=((-5+-sqrt(153))/(8))(x-2)`.