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Samuel Bishop
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1 4 TANGENTS AND NORMALS OBJECTIVE PROBLEMS. For a curve = f(x) if = x then the angle made b the tangent at (,) with OX is ) π / 4 ) π / tan tan / ) 4). The angle made b the tangent line at (,) on the curve = 4x-x with OX is ) ) tan tan ) tan / π 4) 4 8. Slope of the tangent line at x = on = is 4 + x ) / ) -/ ) - 4) None x 4. Slope of the normal line at (a, b) to the curve + = is a b ) /b/a ) b/a )-a/b 4) a/b 5. Slope of the normal line to the curve ) 5/8 ) -8/5 ) - 4) None x x n + = at (,-) is 6. The slope of the normal line to the curve x = a ( t sin t ), = a ( cos t) at a point t is ) tan t/ ) cot t/ ) cot t/ 4) tan t/ 7. Equation of the tangent line at (,4) to the curve ( ) ) x = 0 ) 0x = ) 0x + = 4) none n x = 6x is 9. Equation of the normal line to the curve ( x + ) = 5 at the point (,5/) is ) 7x+5 = ) 7x-5 = ) 5x-7 = 4) none 0. Slope of the tangent to ) - ) ) 0 4) = 4 x at the point where abscissa and ordinate are equal is

2 5. Length of the subnormal at (0,b) to = be x/a is ) / a ) a ) b a 4) / a. A point which the tangent to the curve x ( x ) ) (-,9) ) a ) a / 4) none. A point at which the tangent to the curve x ( x ) ) (-,9)) ) (,-) ) (,) 4) (,) 4 4. Length of the sub tangent at (-a,a) on x a ( a 0) ) a ) a ) a 4) 4a 5. Length of the subtangent at an point on ) Proportional to abscissa ) Proportional to ordinate ) Length of the subnormal 4) None 6. The length of the subtangent at an point of ) Varies as the abscissa ) Varies as the ordinate ) Constant 4) Length of the subnormal 7. Equation of the tangent at (,-) to the curve ) x-=0 ) x+=0 ) -=0 4) +=0 8. The tangent line at = is parallel to the x axis is = is parallel to the x-axis is = > is = is n n a x x/a = be is x x 4x x 5x = is a a, 8 8 to the curve / / / x + = a is parallel to the line ) x = - ) x = ) x = 0 4) = 0 9. If ψ is the angle made b the tangent line at a point P(x, ) on the curve = f(x) with OX then length of the normal is ) secψ ) cosψ ) cosecψ 4) tan ψ 0. At an point on the curve = f(x) if m = then length of the tan gent length of the normal = ) /m ) / m ) m 4) m

3 . At an point on the curve = f(x) if m = then length of thesubnormal length of thesub tan gent = 6 ) Constant ) m ) m 4) / m. If the normal to the curve = f(x) at (,4) makes angle π / 4 with OX then f () = ) - ) ) -/4 4) 4/. The length of subtangent, ordinate of the point and length of subnormal at a point on the curve = f(x) are in ) A.P. ) G.P. ) H.P. 4) None 4. For the parabola ) : ) : ) x: 4) = 4ax the ratio of the length of subtangent to abscissa at an point is x : 5. The length of the normal at an point P(x, ) on the hperbola ) a ) a ) OP where O is the origin 4) None 6. The length of the subtangent at an point on ) a ) a ) b 4) b 7. Length of the subtangent at ( x, ) on n ) x m n ) x m m ) x n n 4) m x/b ae,m,n 0 = > is n m m+ n x a m,n 0 = > is 8. If the gradient to the curve x+ax+b=0 at (,) is then (a,b) = ) (-,) ) (,-) ) (-,) 4) (,) 9. If the tangent line at (x, ) to the curve ) ± ) ± ) 4) None x x 8 x = a is = + is parallel to the X-axis then x = 0. The points on the curve = sinx where the tangent lines are parallel to X- axis are given b ) x = n π,n Z ) x = n π,n Z ) x = ( n + ) n /, n Z 4) None. The distance from the origin to the normal at x = 0 to the curve ) 5 ) / 5 ) / 5 4) x = e + x is

4 . If at a point ( x ), equal, then length of normal is on the curve = f(x) if lengths of sub tangent and subnormal are 7 ) ( / ) ) ) 4). If ax + b + c = 0 is normal to the curve x =, then ) a>0, b<0 ) a>0, b>0 ) a<0,b<0 4) none 4. The curves = x + x + and ) (,) ) )(-,-) ) (0,) 4) (-,-9) 5. If the curves ax ax 4a = x + 5x touch each other at the point = + and = x + 4a touch at x = 0 then a = ) - ) ) -4 4) 4 6. The common tangent line to the parabola touch each other, is ) x-axis ) A bisector of coordinate axes ) -axis 4) None = x and circle x 4x 0 + = at which the 7. Area of the triangle formed b the tangent, normal at (,) on the curve x + = and the x-axis is ) sq. Unit ) sq. Units ) / sq. Units 4)4 sq. Units 8. Area of the triangle formed b the tangent at( x, ) to ) x ) x ) / x 4) a 9. The curves = = cut each other at an angle x x,x ) 0 ) π / 4 ) π / 4) π / 40. The x-intercept made b the tangent at t on the curve ) a ) a sin t ) a cos t 4) a tan t 4. Equation of the normal line at ' ' ) xcos θ / + sin θ / = aθcos θ / + asin θ / ) xcos θ / sin θ / = aθsin θ / ) xsin θ / cos θ / = a sin θ / 4) None x = a and the coordinate axes is = = is x a cos t, a sin t θ to the curve x a ( sin ), a ( cos ) = θ = θ = θ is

5 4. Equation of the normal at x = 0 to the curve ( ) x sin ( sin x) = + + is 8 ) x = ) x + = ) x + = 0 4) x + = 4 If the normal line to the curve x = at ( m, m ) is ) /4 ) 9 / ) /9 4) / = mx 4m then m 44. The -intercept of the tangent at an point of the curve ax b + = is proportional to ) Cube of the abscissa ) Cube of the ordinate ) Square of the ordinate 4) None 45. Length of the tangent to the curve x = a ( cos t + log tan t / ), = a sin t at an point t on it. ) Varies as abscissa ) Varies as ordinate ) Varies as length of normal 4) Constant 46. At the point (a, a) on ) Length of the subnormal x = a x ) Square of the length of S.N ) Twice the subnormal 4) None length of the sub tangent = 47. If the relation between sub-normal SN and sub-tangent ST at an point on the curve b = (x + a) is p(sn) = q(st) then p/q = a) 8/7a b) 8/7 c) 8/7b d) 8b/7 48. The condition that the two curves x =, x = k cut orthogonall is a) k = b) 8k = c) 8k = d) k = 49. The condition that the two curves = 4ax, x = c cut orthogonall is a) c = 6a b) c = a c) c 4 = 6a 4 d) c 4 = a If the tangent at P on the curve x m n = a m + n meets the axes at A and B then AP : PB = a) m : n b) n : m c) : d) :

6 5. If the tangent at the point (at, at ) on the curve a = x meets the curve again at 9 a) at 4 at, 8 b) at 4,8at at c),at d) (at, at) 5. If the tangent to the curve = ax + x at the point (a, a) cuts off intercepts α and β on the coordinate axes, where α + β = 6 then the value of a is. (). (). () 4. (4) a) 6 b) 8 c) 0 d) (,) (,) = = tan ψ TANGENTS AND NORMALS HINTS AND SOLUTIONS = 4 () = = tan ψ where ψ is the angle made b the tangent with OX. 8 x = = x= (4 + x ) x= Differentiating and putting x = a, = b; n n a b n + n = 0 b a b b b = a Slope of normal = a b

7 5. (4) 0 6. (4) 7. () 8. () 9. () 0. () Differentiating and putting x =, = ; ()( ) + ( ) + () + ( )() = 0 = 0 t = a( cos t) and = a(sin t) = cot. dt dt Equation of the tangent line is ()6 (4) 4 = (x ) i.e., 0x + =. Equation of the normal line is 5 5 x = ( ) ( + ) i.e., 7x 5 =. x = and = 4 x x = ± po int = (, ) Slope of tangent = (, ) =. = b e a x / a Length of S.N. at (0, b) = b b b = a a

8 . () = x(x ) + x (x ) = x(x )(x ). Slope = 0 x = 0 or or.. () Taking logarithms and differentiating; + = 0 = x x = x = a. () n n = a x =. nx Length of S.T. = 4. () 5. (4) nx = n x x b a Length of S.T. = a = b = constant x / a b = b e = a a = x = f (x 8x + 5) f ( x x + ) m = 0. (Putting x =, = ) 6. () / / / x + = a / / x + = 0 = x / (a / 8,a / 8) =

9 7. () We have tan ψ = Length of the normal = + tan ψ = sec ψ 8. () 9. () 0. (). (). (). () Length of tangent = = Length of normal ( / ) m Length of S.N. = = m Length of S.T. Slope of tangent = f (x) slope of normal = f (x) = f (x) = π = tan f (x) 4 Length of S.T., ordinate of the point, length of S.N. = /( / ),, are in G.P. with C.R. = m =. = 4ax = 4a a = / = a 4ax S.T.: x = : x = : x = : a a x = and length of normal =

10 x + = x + = OP 4. () Length of S.T. = = b b 5. () 6. () 7. () Taking logarithms and differentiating: n = mx mx m Length of S.T. = = x n n (, ) curve a + b = Differentiating and putting x =, = () + + a + b = 0 (a + ) = = a + b = b + 8. () 9. () 0 x 0 x = = = ±. = 0 cos x = 0 π x = (n + ) where n Z. x = 0, x = e + x =. x e x = +. Equation of normal is = x.

11 0. (4) 4. (). (). () 4. () 5. () / = = at an point (x, ) Length of normal = + m = A point on the curve x = is (t, /t) (t 0) x = = = x x Slope of the normal at t = t = a, b of opposite sign. a > 0 b B substitution we see that the point (, ) lies on both the curves. = x + x + = x + = 4 = x + 5x (,) = (x + 5) = 4 Putting x = 0, = 4a, (,) m = [ax a] r=0 = a, m = m = m a = = x parabola and (x ) + ( 0) = Circle touch each other and touch -axis at the origin. (,) = and area of the triangle

12 + + ( ) ( ) = = = sq.unit () 5 6. () 7. (4) 8. () 9. () Equation of the tangent is x + x = a (x, ) curve x = a. Area of the triangle = x-intercept -intercept a a = x = a = x Let P(x, ) be a point of intersection. x x = x 6x = 0 x = x x = x + 6x = 0 x = x At ever point of intersection product of the slopes =. Equation of tangent is a(sin θ ) θ = = tan a( + cos θ) Equation of the normal line: x + =. a cos t a sin t

13 θ x a( θ + sin θ ) = tan [ a( cos θ )] () 4. () θ θ θ θ i.e., x cos + sin = a cos sin θ + x = 0 = the point is (0, ) sin x cos x = ( + x) log( x) x 4 sin x = = (0,) + 0 Equation of the normal is = (x 0). 4. () = x x m = and = (m, m ) = slope of normal = m m m = 9 a b a + = = x bx (x, ) curve x a b + = Equation of the tangent at a point (x, ) is a bx = (x x ) x i.e., + = + bx a bx a Y-intercept = a a b + =. ab x b

14 4. (4) (4) 45. () 46. () a cos t = t a sin t + sec tan(t / ) cos t = = tan t ( sin t + (/ sin t) Length of tangent = a sin t tan t + tan t = a B log differentiation : = + x a x At the point (a, a): and S.T. a = = a m+ n m n n x = a (m + n) = nx nx n (S.T) = x, (m n) = + m + n (m + n) m + n (S.N.) = = nx n x f ( xe ) f e x x + = = = x x + e x 0 B substitution we find = 0 for the point (, 0). tangent is vertical. 47. (4)

15 48. () 8 f g f g + = 0 x x x = 0 x = / x = k, x = = k and x = 4k = 49. (4) 50. () 5. () 5. ()

y mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent

y mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent Mathematics. The sides AB, BC and CA of ABC have, 4 and 5 interior points respectively on them as shown in the figure. The number of triangles that can be formed using these interior points is () 80 ()