The centre of a circle is O. There is a line XY which shares one common point with the circle. A line segment OP is drawn from O to line XY such that P is a point on line XY. What happens if P lies on the circle?
The centre of a circle is O. There is a line XY which shares one common point with the circle. A line segment OP is drawn from O to line XY such that P is a point on line XY. What happens if P lies on the circle?
The correct option is C OP is perpendicular to XY.
XY touches the circle at only one point. Thus, it is a tangent to the circle. The tangent touches the circle at only one point which lies on the circumference (point P).
Thus, any other point on line XY other than P lies outside the circle. Let such a point be Q. Now let us compare lengths OP and OQ. It is clear that OP is equal to radius since P lies on circle. Q lies outside. Thus, OP is lesser than OQ. A point which is outside the circle will obviously be farther away from circle centre than a point which lies on the circle.
It can be seen that at any position of point Q on line XY except on P, OP is lesser than OQ. In other words OP is smallest distance between O and XY. The smallest distance between a line and a point is the perpendicular distance. Hence OP is perpendicular to XY.
OP is perpendicular to XY.
XY touches the circle at only one point. Thus, it is a tangent to the circle. The tangent touches the circle at only one point which lies on the circumference (point P).
Thus, any other point on line XY other than P lies outside the circle. Let such a point be Q. Now let us compare lengths OP and OQ. It is clear that OP is equal to radius since P lies on circle. Q lies outside. Thus, OP is lesser than OQ. A point which is outside the circle will obviously be farther away from circle centre than a point which lies on the circle.
It can be seen that at any position of point Q on line XY except on P, OP is lesser than OQ. In other words OP is smallest distance between O and XY. The smallest distance between a line and a point is the perpendicular distance. Hence OP is perpendicular to XY.
