William Anthony Granville

“ Solving this equation, we find that or 2, giving points C and D where the curve (or tangent) is parallel to OX.
(d) = 45°, ; therefore . Solving, we get , giving two points where the slope of the curve (or tangent) is unity.
(e) Slope of line ; therefore . Solving, we get , giving points E and F where curve (or tangent) is parallel to line AB.
Since a curve at any point has the same direction as its tangent at that point, the angle between two curves at a common point will be the angle between their tangents at that point. ”

“ Find the equation of the tangent at to the ellipse .
Ans. .
6. Find equations of tangent and normal to the witch as at the point where .
Ans. .
7. Prove that at any point on the catenary the lengths of subnormal and normal are respectively.
8. Find equations of tangent and normal, lengths of subtangent and subnormal, to each of the following curves at the points indicated:
9. Prove that the length of subtangent to is constant and equal to .
10. Get the equation of tangent to the parabola which makes an angle of 45° with the axis of X.
Ans. . ”

“ Find the direction of motion of the projectile
(a) at instant of projection;
(b) when it strikes a vertical cliff miles distant.
(c) Where will the path make an inclination of 45° with the horizontal?
(d) Where will the projectile travel horizontally?
Ans. (a) ; (b) 135°; (c) ; (d) (1, 1) . ”