Common Tangents to Two Conics Two conic sections are said to be tangent if they intersect at exactly one point, and the distance between the two points of i...
Common Tangents to Two Conics
Two conic sections are said to be tangent if they intersect at exactly one point, and the distance between the two points of intersection is equal to the radius of either conic. The tangents to the two conic sections at a point of intersection are called the tangents to the conic sections.
For example, if two circles are tangent to each other, then the tangents to each circle are perpendicular and equal in length. Similarly, if two ellipses are tangent to each other, then the tangents to each ellipse are parallel and equal in length.
The equation of the line that passes through the points of intersection of two tangents to a conic is given by the equation:
where (m) is the slope of the tangent, and (b) is the y-intercept.
The distance between the points of intersection of two tangents to a conic is given by the formula:
where (a_1) and (a_2) are the coefficients of the squared terms in the equations of the two conics.
These formulas can be used to find the equations of the lines that pass through the points of intersection of two tangents to a conic