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Tangent Lines

The tangent line (or simply the tangent) to a curve at a given point is the straight line that just touches the curve at that point. As it goes through the point where the tangent line and the curve meet, or the point of tangency, the tangent line is going in the same direction as the curve, and in this way it is the best straight-line approximation to the curve at that point.

The word tangent is derived from the Latin word "tangere". In Geometry, the tangent line has many applications in conic sections, circles, etc..

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Construction of Tangent to a Circle

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Given below are the steps necessary to draw the circle's tangent line at a given point on it.

Construction :

Tangent to a Circle

  1. First, we have to make a circle for any radius, with O as its centre.
  2. Take any point, P on the circle and join the two points O and P.
  3. With P as centre, we have to draw an arc which cuts the line OP at the point M.
  4. Taking M as the centre and with the same radius, we have to draw an arc that cuts the previous arc at N. Again, taking N as centre and for the same radius, draw an arc that cuts the first arc at L.
  5. Draw the bisector for the angle ∠ NPL that is PT
  6. Then the angle OPT will be equal to 90° and now, PT is the tangent at the point P.

Tangent Lines to Ellipse and Parabola

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Tangent Lines to Ellipse and Parabola

The tangent line equation for ellipse at (x1, y1) is given by `(x x_1)/a^2` +` (yy_1)/b^2` = 1

The tangent line equation for parabola at (x1, y1) is given by 2ax - y1y + 2ax1 = 0

Practice Problems on Tangent Lines

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Problem 1 : Construct geometric tangent line for the circle x^2+y^2 = 16 at the point (1,2)

Problem 2 : Construct geometric tangent line for the curve `(x ^2)/4^2` +` (y^2)/6^2` = 1 ellipse at the point (2.,5)

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