BodhiAI

Nov. 15, 2019

Normal to the Circle at a Given Point

The normal at any point on the circle is a straight line which is perpendicular to the tangent to the curve at that point and always passes through origin (for the circle).

Now, to find the equation of normal, follow the following steps.

Step I Write the equation of the tangent to the circle at the given point P(x_{1} , y_{1}) using Point Form.

Step II Find the equation of normal line perpendicular to tangent in STEP I and also passing through (x_{1} , y_{1}) the equation thus obtained in STEP II is the equation of the normal.

Equation of Pair of Tangents

Let be any point lying outside the circle then the equation of pair of tangents represented by lines PA and PB is

where,

= {which is positive as point lies outside the circle}

=

Equation of Chord of Contact

Let PA and PB be any two tangents to the circle from some external point in the plane of the circle

Then the line segment AB is known as the chord of contact and its equation is given by

Equation of Chord if its Mid Point is Known

Let is the midpoint of chord AB of the circle

then, the equation of chord of circle through this point is given by the equation

Power of a Point

Given a point P and a circle, pass two lines through P that intersect the circle in points A, D and B,C respectively. Then AP.DP = BP.CP

The point P may lie either inside or outside the circle. The line through A and D (or that through B and C or both) may be tangent to the circle, in which case A and D coalesce into a single point. In all the cases, the theorem holds and is known as the Power of a Point Theorem.

When the point P is inside the circle, the theorem is also known as the Theorem of Intersecting Chords. When the point P is outside the circle, the theorem becomes the Theorem of Intersecting Secants.

The proof is exactly the same in all three cases mentioned above. Since triangles ABP and CDP are similar, the following equality holds

which is equivalent to the statement of the theorem: AP.DP = BP.CP.

The common value of the products is then depends only on P and the circle and is known as the Power of Point P with respect to the circle. Note that, when P lies outside the circle, its power equals the length of the square of the tangent from P to the circle.

i.e. PA.PB