The tangent and the normal drawn to the curve y=x²-x+4 at P(1,4) cut the X -axis at A and B respectively. If the length of the subtangent drawn to the curve at P is equal to the length of the subnormal, then the area of the P A B in square units is,

Step-by-step solution

Given, equation of curve is y=x²-x+4 Slope of tangent at P(1,4) is d y d x =2 x-1 ( d y d x )_ (1,4) =2-1=1 So, equation of tangent is y-4=1(x-1) y-x=3...(i) and cquation of normal at point P(1,4) is aligned &y-4=-1(x-1) &x+y=5 ...(ii) aligned Since, the tangent cuts X -axis at A . Therefore, coordinates of A are (-3,0) and the normal cuts X -axis at B and coordinates of B are (5,0) . Therefore, area of P A B= 1 2 | array ccc 1 & 4 & 1 -3 & 0 & 1 5 & 0 & 1 array |= 1 2 [1(0)-4(-3-5)+1(0)]= 1 2 |32|=16 sq. units