(a) The graph of \(y = 2px^{2} - p^{2}x - 14\) passes through the point
(a) The graph of \(y = 2px^{2} - p^{2}x - 14\) passes through the point (3, 10). Find the values of p.
(b) Two lines, \(3y - 2x = 21\) and \(4y + 5x = 5\) intersect at the point Q. Find the coordinates of Q.
Explanation
(a) \(y = 2px^{2} - p^{2}x - 14\)
At point (3, 10), y = 10 when x = 3.
\(\implies 10 = 2p(3^{2}) - p^{2}(3) - 14\)
\(10 = 18p - 3p^{2} - 14\)
\(-10 = 3p^{2} - 18p + 14 \implies 3p^{2} - 18p + 24 = 0\)
\(3p^{2} - 12p - 6p + 24 = 0\)
\((p - 4)(3p - 6) = 0 \implies \text{p = 2 or 4}\).
(b) \(3y - 2x = 21 ... (1)\)
\(4y + 5x = 5 ....(2)\)
Using elimination method, multiply (1) by 4 and (2) by 3.
\((1) \times 4 : 12y - 8x = 84 ... (3)\)
\((2) \times 3 : 12y + 15x = 15 ... (4)\)
Subtracting (3) - (4), we have:
\(-23x = 69 \implies x = -3\)
Putting x = -3 in (1), we have
\(3y - 2(-3) = 3y + 6 = 21\)
\(3y = 15 \implies y = 5\)
Hence, the coordinates of Q are (-3, 5).