Table of Contents
Explain in Detail :Various Conditions of Tangency
Tangency is a condition of a curve and a line where they intersect and the curve is exactly on the line. There are three conditions of tangency: point, line, and curve.
Point tangency is when the curve and line intersect at a single point. Line tangency is when the curve is exactly on the line. Curve tangency is when the curve is close to, but not exactly on, the line.
When Point Lies on the Circle
A point lies on a circle if it is exactly on the circle. For example, the point (3,4) lies on the circle with center (0,0) because 3+0=3 and 4+0=4.
When Point Lies Inside the Circle
If point P lies inside the circle, then the distance from P to the center of the circle is less than the radius of the circle.
When Point Lies Outside the Circle
If point P lies outside the circle, then PQ is not a diameter.
If PQ is not a diameter, then the angle at P is not a right angle.
Properties of Tangent Lines
A tangent line to a curve at a point is a line that touches the curve at that point and is perpendicular to the tangent line at that point.
The equation of a tangent line to a curve at a point is
y = mx + b,
where m is the slope of the tangent line and b is the y-intercept.
General Equation
Here, the list of the tangent to the circle equation is given below:
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- The tangent to a circle equation x2+ y2=a2 at (x1, y1) is xx1+yy1= a2
- The tangent to a circle equation x2+ y2+2gx+2fy+c =0 at (x1, y1) is xx1+yy1+g(x+x1)+f(y +y1)+c =0
- The tangent to a circle equation x2+ y2=a2 at (a cos θ, a sin θ ) is x cos θ+y sin θ= a
- The tangent to a circle equation x2+ y2=a2 for a line y = mx +c is y = mx ± a √[1+ m2]
Condition of Tangency
The tangent is considered only when it touches a curve at a single point or else it is said to be simply a line. Thus, based on the point of tangency and where it lies with respect to the circle, we can define the conditions for tangent as:
- When point lies inside the circle
- When point lies on the circle
- When point lies outside the circle
When the Point Lies Inside the Circle
Consider the point P inside the circle in the above figure; all the lines through P is intersecting the circle at two points.
It can be concluded that no tangent can be drawn to a circle which passes through a point lying inside the circle.
When the Point Lies on the Circle
From the figure; it can be concluded that there is only one tangent to a circle through a point which lies on the circle.
When the Point Lies Outside the Circle
From the above figure, we can say that
There are exactly two tangents to circle from a point which lies outside the circle.
Tangent Properties
- The tangent always touches the circle at a single point.
- It is perpendicular to the radius of the circle at the point of tangency
- It never intersects the circle at two points.
- The length of tangents from an external point to a circle are equal.
Tangent Formula
Suppose a point P lies outside the circle. From that point P, we can draw two tangents to the circle meeting at point A and B. Now let a secant is drawn from P to intersect the circle at Q and R. PS is the tangent line from point P to S. Now, the formula for tangent and secant of the circle could be given as:
PR/PS = PS/PQ
Tangent Theorems
Theorem 1: The tangent to the circle is perpendicular to the radius of the circle at the point of contact.
Theorem 2: If two tangents are drawn from an external point of the circle, then they are of equal lengths
