EXAMPLES IX 141 



A point of inflexion occurs when -j-2 = o 



That is, when 12a^ + I2x - 72 = 



or x 2 + x - 6 = 



(x + 8)(x - 2) = 



at points where x = 3 and x = 2 



EXAMPLES IX 



(1) Find the values of the slope of the curve y = a? 8x + 5 

 at the points where x = 1-5 and x = 2-0. Find the equations 

 of the tangents to the curve at these points. What is the angle 

 between these tangents ? 



(2) Find the value of the slope of the curve y = 3x 2 4c + 3 

 at the point where x = 2. Find the equations of the tangent and 

 normal to the curve at that point. 



(3) The curve y = x 2 1 is cut by the line y = x + 5. Find the 

 co-ordinates of the points of intersection. Find the angles be- 

 tween the line and the curve at these points. 



(4) Find the equations of the tangent and normal to the curve 

 y = 4r at the point where x = 2. 



(5) Find the equations of the tangent and normal to the curve 

 t/ 3 = &r 2 at the point where x =2. 



(6) Find the co-ordinates of the point of intersection of the 

 curves x 2 + y z = 5 and x 2 y 2 = 2, and find the angle between the 

 curves at that point. 



(7) The two curves xy = 1 and x 2 y 2 = 4 intersect at a point P. 

 Find the co-ordinates of P and the angle between the two curves 

 at that point. 



(8) The curve xy = 4 is cut by the line IQy = 7x + 4. Find the 

 co-ordinates of the points of intersection and the angles between 

 the line and the curve at these points. 



(9) The curve y = ox n passes through the points (3, 10) and 

 (6, 17). Find a and n. Find the value of the slope of the curve 

 at a point P where a? = 2. A second point Q is taken on the 

 curve, and this point can be on either side of P. Find the 

 co-ordinates of the two positions of Q, so that the angle turned 

 through in moving along the curve from P to Q is 7. 



(10) The two curves y z = Sx and x 2 = 8y intersect at a point P, 

 other than the origin. Find the co-ordinates of P and the angle 

 between the curves at that point. 



(11) The curve y = ad** passes through the points (1, 8-5) and 

 (10, 12-6). Find a and b. Find the value of the slope of the 

 curve at the point where x = 5. Find the equations of the tangent 

 and normal to the curve at that point. 



