(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 \(4y + 5x = 5\) i...
Question 1 Report
(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.
Answer Details
(a) Since the graph passes through \((3,10)\), substitute \(x=3\) and \(y=10\) into \(y=2px^2-p^2x-14\):
\[10=2p(3)^2-p^2(3)-14.\]
\[10=18p-3p^2-14\]
\[3p^2-18p+24=0\]
\[p^2-6p+8=0\]
\[(p-2)(p-4)=0.\]
Therefore,
\[\boxed{p=2\text{ or }p=4}.\]
For these two values, the corresponding curves are \(y=4x^2-4x-14\) and \(y=8x^2-16x-14\). Both pass through \((3,10)\), as shown below.
(b) The equations of the two lines are
\[3y-2x=21 \quad \text{and} \quad 4y+5x=5.\]
Multiply the first equation by 5 and the second equation by 2:
\[15y-10x=105\]
\[8y+10x=10.\]
Adding the equations gives
\[23y=115\]
\[y=5.\]
Substitute \(y=5\) into \(3y-2x=21\):
\[3(5)-2x=21\]
\[15-2x=21\]
\[-2x=6\]
\[x=-3.\]
Hence, the coordinates of \(Q\) are
\[\boxed{Q=(-3,5)}.\]
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