A straight line passes through the point P(-1, 3). Another line which passes through Q(-4, 4) intersects the first line at the point R(k, 5), where k is a constant. If \(<PRQ = 90°\), find the values of k.
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Correct Answer: Option
Explanation:
\(< PRQ = 90°\)
Gradient of PR \(\times\) Gradient of QR = -1.
\(\frac{5 - 3}{k + 1} \times \frac{5 - 4}{k + 4} = -1\)
\(\frac{2}{(k + 1)(k + 4)} = -1\)
\(-2 = k^{2} + 5k + 4 \implies k^{2} + 5k + 4 + 2 = 0\)
\(k^{2} + 5k + 6 = 0\)
\(k^{2} + 3k + 2k + 6 = 0\)
\(k(k + 3) + 2(k + 3) = 0 \implies (k + 3)(k + 2) = 0\)
\(\implies \text{k = -2 or -3\)
\(< PRQ = 90°\)
Gradient of PR \(\times\) Gradient of QR = -1.
\(\frac{5 - 3}{k + 1} \times \frac{5 - 4}{k + 4} = -1\)
\(\frac{2}{(k + 1)(k + 4)} = -1\)
\(-2 = k^{2} + 5k + 4 \implies k^{2} + 5k + 4 + 2 = 0\)
\(k^{2} + 5k + 6 = 0\)
\(k^{2} + 3k + 2k + 6 = 0\)
\(k(k + 3) + 2(k + 3) = 0 \implies (k + 3)(k + 2) = 0\)
\(\implies \text{k = -2 or -3\)