if secAand cosecA are roots of x^2-px+q then choose the correct option

a)p^2+q^2=2q

b)p^2+q^2=2p

c)p^2-q^2=2q

d)p^2-q^2=2p

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Solution

x²- px + q Here a =1, b =-p c = q Sum of roots secA + cosecA =-b/a = p.......1) And product of the roots secA.cosecA = c/a = q.......2) From 2) 1/(cosA.sinA) = q or sinAcosA = 1/q...........3) Dividing 1) by 2) we have (secA + cosecA)/(secA.cosecA) = p/q 1/cosecA + 1/secA = p/q sinA + cosA = p/q Squaring both sides, we have Sin²A + cos²A + 2sinA.cosA = p²/q² (Sin²A + cos²A) + 2(sinA.cosA) = p²/q² 1 + 2/q = p²/q² [from 3) sinAcosA = 1/q ] P² = q²(1 + 2/q) P² = q² + 2q P²-q² = 2q Ans. Correct option C

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