Question

For a given circle, while constructing a tangent to it through a point on it, which property of a tangent is used if you make use of the centre of the circle?

Answer 1

Hint: To solve the given question, we will first find out what a circle is and what a tangent to a circle is. Then we will make use of the fact that the line joining the centre of the circle to that point on the circle where we are going to draw the tangent will be perpendicular to the tangent drawn at that point. With the help of this concept, we will explain the required property which we will use if we have to draw a tangent on a point (x, y) on the circle and the centre (a, b) of the circle is given.

Complete step-by-step answer:
Before we solve the given question, we must know what a circle is and what a tangent to a circle is. A circle is a locus of a point whose distance from a fixed point always remains constant. A tangent to a circle is a line which touches the circle at only one point. Now, the rough sketch of the circle with a tangent is shown below.

In the above figure, O is the centre of the circle and P is the point on the circle where the tangent AB is drawn. Now, OP is normal to the circle and it is perpendicular to the tangent AB. Also OP = radius. Thus, the property which we are going to use is “The perpendicular distance of any tangent from the centre will be equal to the radius of that circle”. By the help of this property, we will derive the equation of the tangent.

Note: For using this property, we must know the coordinates of the centre and the coordinates of the point at which we are going to draw a tangent. If we do not know the circle’s centre, we must know at least the radius of the circle and the point at which we are drawing the tangent.