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Tangent from an External Point


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

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

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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