Theory question 111
Matrices and Determinants(a) Given that |5 2 −3; −1 k 6; 3 9 (k + 2)| = −207, find the values of the constant k.
(b) The equation of a curve is x(y2 + 1) − y(x2 + 1) + 4 = 0. Find the (i) gradient of the curve at any point (x, y); (ii) equation of the tangent to the curve at the point (−1, −3)
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(a) 5[k(k + 2) − 54] − 2[−(k + 2) − 18] − 3[−9 − 3k] = −207
5k² + 10k − 270 + 2k + 40 + 27 + 9k = −207
5k² + 21k + 4 = 0, (5k + 1)(k + 4) = 0
k = −1/5 or k = −4
(b)(i) Differentiate implicitly: (y² + 1) + 2xy dy/dx − (x² + 1)dy/dx − 2xy = 0
dy/dx = (2xy − y² − 1)/(2xy − x² − 1)
(ii) At (−1, −3): dy/dx = (6 − 9 − 1)/(6 − 1 − 1) = −4/4 = −1
Tangent: y + 3 = −(x + 1), i.e. x + y + 4 = 0
Expand the determinant along the first row; in (b) differentiate each term with respect to x, treating y as a function of x.