Home
Class 12
MATHS
If tanthetaa n dsectheta are the roots o...

If `tanthetaa n dsectheta` are the roots of `a x^2+b x+c=0,` then prove that `a^4=b k^2(4a c-b^2)dot`

Text Solution

Verified by Experts

`tan theta + sec theta = - (b)/(a)` (1)
` tan theta sec theta = (c)/(a)` (2)
Now, `sec^(2) theta - tan^(2) theta = 1`
`rArr sec theta - tan theta = - (a)/(b) ` (3)
From (1) and (3) .
`rArr sec theta = - ((a^(2) + b^(2)))/(2ab) and tan theta = ((a^(2) - b^(2)))/(2ab)`
substituting these values in Eq. (2), we have
`((a^(2) + b^(2))(b^(2) - a^(2)))/(4a^(2)b^(2)) = (c)/(a)`
or `b^(4) - a^(4) = 4acb ^(2)`
or `a^(2) = b^(2)(b^(2) - 4ac)`
Promotional Banner

Topper's Solved these Questions

  • THEORY OF EQUATIONS

    CENGAGE|Exercise Exercise 2.10|5 Videos

  • THEORY OF EQUATIONS

    CENGAGE|Exercise Exercise 2.11|8 Videos

  • THEORY OF EQUATIONS

    CENGAGE|Exercise Exercise 2.8|11 Videos

  • STRAIGHT LINES

    CENGAGE|Exercise JEE Advanced Previous Year|4 Videos

  • THREE DIMENSIONAL GEOMETRY

    CENGAGE|Exercise Question Bank|20 Videos

Similar Questions

Explore conceptually related problems

If beta+cos^(2)alpha,beta+sin^(2)alpha are roots of x^(2)+2bx+c=0 and gamma+cos^(4)alpha,gamma+sin^(4)alpha are the roots of X^(2)+2BX+C=0 then prove that b^(2)-B^(2)=c-C

If roots of equation a-b)x^(2)+(b-c)x+(c-a)=0 are equal then prove that b+c=2a

If roots of ax^(2)+bx+c=0 are alpha,beta and roots of Ax^(2)+Bx+C=0 are alpha+k,beta+k .then prove that ((B^(2)-4AC))/((b^(2)-4ac))=(A/a)^(2)

if ax^(2)+bx+c=0 has imaginary roots and a+c

If a+b+c=0 prove that (a^4+b^4+c^4)/(a^2b^2+b^2c^2+c^2a^2)=2

If alphaa n dbeta,alphaa n dgamma,alphaa n ddelta are the roots of the equations a x^2+2b x+c=0,2b x^2+c x+a=0a d nc x^2+a x+2b=0, respectively, where a, b, and c are positive real numbers, then alpha+alpha^2= a.a b c b. a+2b+c c. -1 d. 0

If a and b are roots of the equation x^2+a x+b=0 , then a+b= (a) 1 (b) 2 (c) -2 (d) -1

If alpha,beta,gamma,delta are the roots of the equation x^(4)+Ax^(3)+Bx^(2)+Cx+D=0 such that alpha beta gamma delta =k and A,B,C,D are the roots of x^(4)-2x^(2)+4x^(2)+6x-21=0 such that A+B=0 then the value of (C)/(A), is