University:
University of Cambridge
Course:
0580 IGCSE | Mathematics
Academic year:

2023
Views:

291

Pages:

4
Author:

Shahbaz A.

Q30 Ex 3.7 page 154 THOMAS’ Calculus Twelfth Edition shahbaz ahmed August 2024 Verify that the given point is on the curve and find the lines that are (a) tangent and(b) normal to the curve at the given point. x2 + y 2 = 25, (3, −4) Solution x2 + y 2 = 25 Putting x=3,y=-4 in equation (1) 32 + (−4)2 = 25 9 + 16 = 25 1 (1) 25 = 25 L.H.S = R.H.S Hence the point (3,-4) is on the curve Now. d 2 d (x + y 2 ) = (25) dx dx d 2 d (x ) + (y 2 ) = 0 dx dx d dy 2x + (y 2 ) =0 dy dx 2x + 2y dy =0 dx dy = −x dx dy x =− dx y y Putting x=3,y=-4 dy 3 = −[ ] dx −4 2 dy 3 = =m dx 4 Where m is the slope of tangent at the point (3,-4)Hence equation of tangent 3 y − (−4) = (x − 3) 4 3 y + 4 = (x − 3) 4 3 9 y+4= x− 4 4 3 9 y = x− −4 4 4 3 25 y = x− 4 4 Let m1 be the slope of the normal. Then mm1 = −1 3 4 m1 = −1 m1 = − 34 Equation of normal at the point (3,-4) with slope m1 = − 43 y − (−4) = − 43 (x − 3) y + 4 = − 43 x + 4 3 y = − 43 x + 4 − 4 y = − 43 x The figure below shows the tangent and normal at the point (3,-4) 4