Question

How can I tell if a line is a tangent to a circle?

Answer 1

Hint: To prove the above statement we need to form a triangle with a circle and a line. Now find the slopes of the lines which are joined to the center of the circle. If the resultant slope satisfies the statement about the perpendicular lines then the line is tangent to the circle.

Complete step by step answer:
Here we need to say if a line is a tangent to a circle.
This given statement is dependent on the data given in the question.
As we have no data in the question. We can answer the statement in a general form.
To answer the question we can use the statement that a line tangent to a circle at a particular point will be perpendicular to the radius line from the center of the circle to the particular point.
Or else we can write the same statement as the radius line of the circle will intersect the circle at a particular point perpendicularly.
As we work about any mathematics on the coordinate plane, all the values will be positive.
Let’s assume the value of the center of the circle be C and the point which touches the circle be P.
Let’s take another point on the line which touches the circle to make a triangle. Assume Q as the other point on the line.
Now let’s work on the formed triangle.
We need to find the slopes of both lines which are connected to the center of the circle that is CP and the line PQ.
Here we can use the statement, that in a coordinate plane if two lines are perpendicular if and only if the slopes of the two lines are having negative reciprocals of each other to find the slopes.
If the resultant slopes satisfy the statement then the line which touches the circle will be tangent to the circle.

Note:
 As we know that the tangent line to circle touches only one particular point making $ {{90}^{\circ }} $ with the circle radius line but the tangent line always lies outside of the circle and doesn’t intersect or enter the circle at any other point.