RVHS H2 Math Differentiation Tangents & Normals Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Tangents and Normals 1 Differentiation: Tangents and Normals 1. AJC/2009/I/7 (a) Find 1 2 d1 cosdxx −    and simplify your answer. [2] (b) The curve C has parametric equations ( )ln sinxt= , cotyt= , where 0 t  . Show that d2 d sin 2 y xt=− . [2] P is the point on the curve C where the normal is parallel to the line 22yx=− . Find the equation of the normal at P. [2] This normal intersects the x-axis at the point Q. R is a point on the curve C such PQ = RQ. Find the equation of the normal to the curve C at the point R. [2] 2. AJC/2013/II/4 A curve C has parametric equations ( ) 11 tanxt t −=+ , ( ) 11 tanyt t −=− where 𝑡 ∈ ℝ, t  0. The curve has an oblique asymptote y = ax. (i) Find d d y x in terms of t and show that a = 1. [3] (ii) Sketch C, showing clearly all the asymptote(s), axial intercepts and end points. [3] Hence find the range of values of k such that the line y kx= does not intersect C. [1] (iii) Find the equation of the normal to C at the point where t =1. Hence find the value of the parameter t at the point where the normal intersects C again. [3] 3. RI/2010/I/10 The curve C has equation e−= xyx for 0x and ( , e ) aP a a − is a point on C . (i) Sketch the curve C . [1] (ii) Find, in terms of a , the equation of the tangent to the curve at P . [2] This tangent cuts the y− axis at the point (0, )Qh . Using differentiation, find, as a varies, the exact maximum value of h . [6]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Tangents and Normals 2 4. IJC/2011/II/3 The curve C has parametric equations , where a is a positive constant and . The point P on the curve has parameter . The tangent at P meets the curve again at the point Q. (i) Find in terms of t. [2] (ii) Show that the tangent at P has equation 4 15 3y x a=− . [3] (iii) Find the coordinates of Q. [3] (iv) By considering what happens to x and y for large values of t, find the equation of the asymptote of the curve C. Sketch the curve C. [2] (v) Find, by a non -calculator method, the area of the region bounded by the curve C, the tangent at P and the x-axis. [4] 5. NYJC/2011/I/4 The parametric equations of a curve C are sin 2x a t= and cosy a t= , where a is a positive constant and 22 t−   . (i) Find the equation of the tangent to the curve at the point P where 4t = . [3] (ii) The normal to the curve at the point Q where 3t = intersects the x-axis at R. Find the coordinates of R and hence show that the area enclosed by the normal at Q, the tangent at P and the x-axis is 2 43 3 3 48 2a  − . [5] 6. ACJC/2011/I/12 A curve with equation ( )xy f= is also defined by the parametric equations 21 e , 1x y t t= − = +− , 𝑡 ∈ ℝ. (i) The point P on the curve has parameter p. Given that the tangent to the curve at P passes through the point (1, 0), find the value of p. (ii) Given that the graph of ( )xy f= has an inflexion point at 1t=− , sketch, on separate diagrams, the graphs of ( )xy f= and ( )fyx = , showing clearly the exact coordinates of the turning point(s) and asymptote(s), if any. 2 111,x a y a t t t    = + = −       0t  1 2t =− d d y x

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Tangents and Normals 3 7. NYJC/2011/I/9 The diagram shows a sketch of the curve e xyx −= . (i) By differentiation, find the range of values of x for which the graph of e xyx −= is decreasing. [3] (ii) Determine the range of values of x for which the graph of e xyx −= is decreasing and concave downwards. [3] (iii) The tangent to the curve at P ( ),xy meets the y-axis at R ( )0, h . Express h in terms of x , and find the greatest possible value of h. [6] 8. PJC/2011/I/7 A curve is defined by the parametric equations 3 ,aaxy tt== where a is a constant. (i) Find the equations of the tangent and the normal at point P where 1 2t = . [4] (ii) Find the coordinates of the point where the tangent cuts the curve again. [2] (iii) The tangent at P meets the x-axis at Q and the normal at P meets the x-axis at R. Show that the area of triangle PQR is 2145 6 a . [2] 9. IJC/2012/I/4 The equation of a curve is given by 222 4 66xy y x− + = . (i) Find the exact coordinates of the points on the curve where the tangent is parallel to the y-axis. [4] (ii) Show that every line parallel to the x-axis cuts the curve at two distinct points. [3] y x

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Tangents and Normals 4 10. NYJC/2012/I/3 The equation of a curve is given by 3 2 34 3 2x x y y+ = − . Find d d y x in terms of x and y, simplifying your answer. [2] The curve meets the line yx=− at point P. Find (i) the coordinates of P and [2] (ii) the equation of the tangent at P. [2] The tangent to the curve at P cuts the x-axis at R and the normal to the curve at P cuts the y-axis at Q. If O denotes the origin, use the results above to give a geometrical description of the quadrilateral OQPR. [1] 11. EJC Prelim 9758/2018/01/Q3[Modified] The parametric equations of curve C, is given as 3,x at y at== , where a is a positive constant. (i) The point P on the curve has parameter p and the tangent to the curve at point P cuts the y- axis at S and the x-axis at T. The point M is the midpoint of ST. Find a Cartesian equation of the curve traced by M as p varies. [5] (ii) Find the exact area of triangle OPS, giving your answer in terms of a and p. [2] 12. YJC/2013/I/7 The equation of a curve is 2 1y xy− =− . (i) Find the equations of all tangents to the curve that are parallel to the y-axis. [4] (ii) State and justify whether the curve has any stationary points. [2] (iii) Find the area of the region bounded by the axes, and the normal to the curve at the point where the y-coordinate is 2. [5] 13. SRJC/2013/I/10[Modified] (a) Find the coordinates of the point(s) to the curve 2229x xy y+ − = at which the tangent is parallel to the y – axis. [5] (b) The curve C has parametric equations 2 , 4 .x t t y t= + = − (i) The point P on the curve has parameter p. Show that the equation of the tangent at P is 2(2 1)(4 )p p y x p p+ − − = − − . [2] (ii) Hence, show that every tangent to the curve C does not meet the curve again. [3] (iii) Find the acute angle the tangent at P makes with the x axis when p = 1. [1]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Tangents and Normals 5 14. DHS/2014/I/11 A curve has parametric equations tan , 2cos , for 0 . 2x t y t t = =   (i) Find d .d y x What can be said about the tangent to the curve as 0?t → Hence sketch the curve. [4] (ii) The point P on the curve has a non-zero parameter p. Given that the normal to the curve at P passes through the origin, find the value of p. Hence show that equation of the normal to the curve which passes through the origin is given by 2.yx= [5] (iii) Find the exact area of the region bounded