Choose correct alternatives: If the equation 3x2 – 8xy + qy2 + 2x + 14y + p = 1 represents a pair of perpendicular lines, then the values of p and q are respectively ______.

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KollegeApply · Accepted Answer

If the equation 3x2 – 8xy + qy2 + 2x + 14y + p = 1 represents a pair of perpendicular lines, then the values of p and q are respectively –7 and –3. Explanation: 3x2 − 8xy + qy2 + 2x + 14y + p = 1 Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C = 0 3x2 − 8xy + qy2 + 2x + 14y + p = 1 3x2 − 8xy + qy2 + 2x + 14y + (p − 1) = 0 So the coefficients are: • A=3 • 2H = −8 ⇒ H = −4 • B = q • 2G = 2 ⇒ G = 1 • 2F = 14 ⇒ F = 7 • C = p − 1 A + B = 0 3 + q = 0 ⇒ q = −3 `|(A,H,G),(H,B,F),(G,F,C)| = 0 |(3,-4,1),(-4,-3,7),(1,7,p-1)| = 0 = 3 |(-3,7),(7,p-1)| -(-4)|(-4,7),(1,p-1)| +1|(-4,-3),(1,7)|` 3((−3) (p−1) − (7) (7)) = 3[−3 (p−1) −49] = 3[−3p + 3 − 49] = 3[−3p − 46] = −9p − 138 +4((−4) (p−1) − (7) (1)) = 4[−4(p − 1) −7] = 4[−4p + 4 − 7] = 4[−4p − 3] = −16p − 12 +1((−4) (7) − (−3) (1)) = −28 + 3 = −25 −9p − 138 − 16p − 12 − 25 = −25p − 175 = 0 ⇒ p = −7 p = −7, q = −3

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