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Equation of the Tangent in Point Form is considered one of the most asked concept.
68 Questions around this concept.
Equation of the tangent to the circle, at the point (1, −1), whose centre is the point of intersection of the straight lines x − y = 1 and 2x + y = 3 is :
The range of values of for which the circle
and
have two common tangents is
The number of common tangents that can be drawn to the circle is
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Equation of the Tangent in Point Form
Point Form
The equation of the tangent to a circle at the point P(x1,y1) is
Proof:
NOTE:
In order to find out the equation of a tangent to any 2nd-degree curve, the following points must be kept in mind:
and c will remain c.
This method is applicable only for a 2nd degree conic.
Equation of Tangent of Circle in Parametric Form
The equation of the tangent at the point to a circle
is
Proof:
Slope Form
The equation of the tangent to a circle having slope m is
, and point of tangency is
.
Point of Contact:
NOTE:
Equation of the Normal to a Circle
A line passing through a point P on the curve which is perpendicular to the tangent at P is called the normal to the curve at P.
For a circle, the normal always passes through the centre of the circle.
Point Form:
The equation of the Normal at the point P(x1,y1) to a circle is
Proof:
As we know that the normal always passes through the centre C(-g, -f) of a circle.
Thus, the equation of the normal at point P to the circle
Tangent from a Point to the Circle
If a point lies outside of a circle (here point is P), then two tangents can be drawn from P to the circle. Here, PQ and PR are two tangents.
If a point lies on the circle, then one tangent can be drawn from the point to the circle. If C is the point, then ACB is the tangent
If a point lies inside the circle, then no tangent can be drawn from the point to the circle.
To get equation of the tangents from an external point
The tangents are real, imaginary or coincidence that is depends on the value of the discriminant.
If we have real values of m, then we can find the equations of 2 tangents using these slopes and the point P.
Length of tangent (PT) from a point to a circle
This expression can also be written as
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