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P3 June 2003 A curve has equation 7 x^2 + 48 xy โ 7 y^2 + 75 = 0. A and B are two distinct points on the curve. At each of these points the gradient of the curve is equal to 2 (^11). ( a ) Use implicit differentiation to show that x + 2 y = 0 at the points A and B. (5) ( b ) Find the coordinates of the points A and B. (4) P3 January 2004 The curve C has equation 5 x^2 + 2 xy ๏ญ 3 y^2 + 3 = 0. The point P on the curve C has coordinates (1, 2). ( a ) Find the gradient of the curve at P. (5) ( b ) Find the equation of the normal to the curve C at P , in the form y = ax + b , where a and b are constants. (3) P3 June 2004 The circle C has centre (5, 13) and touches the x -axis. ( a ) Find an equation of C in terms of x and y. (2) ( b ) Find an equation of the tangent to C at the point (10, 1), giving your answer in the form ay + bx + c = 0, where a , b and c are integers. (5) C4 June 2005 A curve has equation x^2 + 2 xy โ 3 y^2 + 16 = 0. Find the coordinates of the points on the curve where dy dx (^) = 0.
C4 January 2006 A curve C is described by the equation 3 x^2 + 4 y^2 โ 2 x + 6 xy โ 5 = 0. Find an equation of the tangent to C at the point (1, โ2), giving your answer in the form ax + by
- c = 0, where a , b and c are integers. (7) C4 June 2006 A curve C is described by the equation 3 x^2 โ 2 y^2 + 2 x โ 3 y + 5 = 0. Find an equation of the normal to C at the point (0, 1), giving your answer in the form ax + by + c = 0, where a , b and c are integers. (7) C4 January 2007 A set of curves is given by the equation sin x + cos y = 0.5. ( a ) Use implicit differentiation to find an expression for dy dx (^). (2)
For โ ๏ฐ < x < ๏ฐ and โ ๏ฐ < y < ๏ฐ,
( b ) find the coordinates of the points where dy dx (^) = 0. (5) C4 January 2008 A curve is described by the equation x^3 ๏ญ 4 y^2 = 12 xy. ( a ) Find the coordinates of the two points on the curve where x = โ8. (3) ( b ) Find the gradient of the curve at each of these points.
C4 January 2010 The curve C has equation cos 2 x + cos 3 y = 1, โ ฯ (^4) ๏ฃ x ๏ฃ ฯ (^4) , 0 ๏ฃ y ๏ฃ ฯ (^6). ( a ) Find
dy
dx in terms of x and y.
The point P lies on C where x =
( b ) Find the value of y at P. (3)
( c ) Find the equation of the tangent to C at P , giving your answer in the form ax + by + c ๏ฐ = 0,
where a , b and c are integers. (3) C4 June 2010 A curve C has equation 2 x^ + y^2 = 2 xy. Find the exact value of
dy
dx at the point on C with coordinates (3, 2).
C4 June 2011 Find the gradient of the curve with equation ln y = 2 x ln x , x > 0, y > 0, at the point on the curve where x = 2. Give your answer as an exact value. (7) C4 January 2012 The curve C has the equation 2 x + 3 y^2 + 3 x^2 y = 4 x^2. The point P on the curve has coordinates (โ1, 1). ( a ) Find the gradient of the curve at P.
( b ) Hence find the equation of the normal to C at P , giving your answer in the form ax + by + c = 0, where a , b and c are integers. (3) C4 June 2012 The curve C has equation 16 y^3 + 9 x^2 y โ 54 x = 0. ( a ) Find dy dx (^) in terms of x and y. (5) ( b ) Find the coordinates of the points on C where
dy
dx = 0.
C4 January 2012 ( a )
(^9) ( b ) 9 x โ 4 y + 13 = 0 C4 June 2012 ( a )
18 โ 6 xy
16 y
2
+ 3 x
2 ( b ) (2, 3 (^2) ), (โ2, โ 3 (^2) )
