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Tangent from an External Point

91345.png
 

Given a point P(a, b) which does not lie on the curve y = f(x), then the equation of possible tangents to the curve y = f(x), passing through (a, b) can be found by solving for the point of contact Q.
 
Let point Q be (x1, y1). Since Q lies on the curve y1 = f(x1).
 
Also, slope of PQ = 91339.png
91333.png.

Condition for which given line is tangent or normal to the given curve

91327.png
 
Let the point on the curve be P(x1, y1) where the line touches the curve. Then P lies on the curve
y1 = f(x1)  ...(1)
 
Also P lies on the line
ax1 + by1 + c = 0  ...(2)
 
Further slope of line = slope of tangent to the curve at point P.
 
91321.png  ...(3)
 
Eliminating x1 and y1 from the above three equations we get the required condition.





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