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5.1. Definition of Tangent, , Let P be a given point on the curve y = f(x). Let Q (x + &x, y + dy) by any, other point on the curve in the neighbourhood of P. Then the limiting position, of the chord PQ when the point Q, travelling along the curve approaches P, is, called the tangent to the curve at the point P., , ¥, , , , Q(x 5x, y+ ay), , Fig. 5.1, , 5.2. Equation of the Tangent, , To find the equation of the tangent at any point P (x, y) of the curve, y=S (), The curve is y=f(x). Let in the neighbourhood P there is point, OQ (x+&x, y+ Sy)., , Equation of the chord PQ is,, _y = 2+d)-y¥ py, ¥-y= (x + &x) - x (X-), = Y-y= ee (X -x),, , where X, Y are taken as current coordinates., , Now as Q — P; then 30,252 and the chord PQ becomes the, , tangent to the curve at P., Therefore the equation of the tangent at the point P (x, y) of the curve, y=f@)is,, , Y-y= a (X - x). Wd)
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COROLLARY 1. If the equation of the curve is given by, , f(x,y) = 0,, then we know that, dy _ _ offox, de —s Of/ay, Hence, the equation of the tangent becomes, wy = — PX py, Y-y= af/ay (X-x), > w-n ot + «x-» =0 wee (2), , COROLLARY 2. If the equation of the curve is given in the parametric form, say, , x=f, y=),, , then we have,, dy _ dy/di _ 9 (), , dx dx/dt ff’ (, Hence, the equation of the tangent is,, , Y¥-o()= fs {x-f(0) ie), , Note : This is known as the equation of the tangent at the point ‘1’ of the curve., 5.3. Geometrical Meaning of dy/dx, The equation of the tangent is,, , Y-y= 2 x-2, , => y=2.x+4 yx, , Also, the equation of a straight line which makes an angle ‘Y with OX and, cuts an intercept c on OY is given by,, Y = mX+c,
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where m = tan '¥,, If we compare these two equations, we get, , 2 = m = tan ¥ = slope of the tangent at (x, y)., , dy, dx, , is equal to the trigonometrical tangent of the angle which the tangent to the, curve at (x, y), makes with the positive direction of the axis of x.”, , Hence, ‘‘the differential coefficient — at the point (x, y) of a curve y =f (x), , COROLLARY 1. If the tangent is parallel to X-axis, then ¥ = 0., , dy, 7 tan '¥ =tan0 = 0., , COROLLARY 2. If the tangent is perpendicular to X-axis then 9 = 90°., , ® = tan = tan 90° = ©, , dx, or aw __1 _1ig, dy dy/dx, , 5.4. Definition of Normal, , The normal to a curve at any point P is a straight line which passes through, the point of contact P and is at right angle to the tangent at P., , 5.5. Equation of Normal, Tangent at point P (x, y) is,, y-y=2a-»
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If m is the gradient of the normal, then, , , , , , me =-l,, dx, ie. m=-1 !, — (dy/dx), «. Equation of the normal at the point P (x, y) is,, 1, , Y= ~ Gyan %~), , > (Y-y) 4 K-y=0, , COROLLARY : If the equation of the curve is given in the parametric form, say, , x=f(), = 9)., then we have, dy _ ¥(), dxf’ (1), , Hence, the equation of the normal is,, , Y-0() =- 0 Ix-s(}, , in, = ¥- 90 =- Gy £0 ix_ 500),, Note : This is the equation of the normal at point ‘r’ of the curve., , 5.6. Angle of the Intersection of Two Curves, , The angle of intersection of two curves is the angle between the tangents to, the two curves at their point of intersection., Let us consider two curves C, and C; intersecting at the point P., , Let m, and mz be the slopes of the tangents at the point P to the two curves, , respectively., Let @ be the angle between the tangents at P. Then,, tan = =, 1 +m, m2, , > 6 = tan 1, 1+m,m,|"
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Notes :, 1. The other angle of intersection is x - 0., , 2. The curves C; and C2 are said to cut orthogonally, if, , x, O=5, , , , Fig. 5.3, Le, tan6 = tan =o, ie, 1+ mm; = 0, Le. mm, = - 1., , 3. The curve C; and C; will touch each other if, m = Mm).
