question_answer 1)

For the curve y = 4x3 – 2x5, find all the points at which the tangent passes through the origin. 

Answer:

Given curve is             y = 4x3 ? 2x5                  ? (1)       Let P(h, k) be a point on (1)          K = 4h3 ? 2h5                                  ?.(2) (1)       m = slope of tangent to (1)             = 12h2 ? 10h4        Equation of tangent to (1) at P(h, k) is             y ? K = m (h ? h)       y ? K = (12h2 ? 10h2) (x ? h) ?. (3)                (3) Passes through (0, 0)        0 ? K = (12h2 ? 10h4) (0 ? h)        K = 12h3 ? 10h5                      ?.. (4)       (2) and (3)             4h3 ? 2h5 = 12h3 ? 10h5       8h5 ? 8h3 = 0          h = 0,  ?1,  1       (2)  K = 0 when k = 0       K = ?4 + 2 = ? 2 when h       = ?1       K = 4 ? 2 = 2 when h = 1       Required points are (0, 0), (?1, ?2) and (1, 2).