Factorise the following: `p^2+6p+8`

Answer

a rarr r; b rarr p; c rarr s; d rarr q`

Explanation

(a) `"Im"(z^(2))=3`
`"or Im"((x+iy)^(2))=3`
`"or "2xy=3`
which is a rectangular hyperbola having eccentricity `sqrt2`.

`tan30^(@)=(b^(2)//a)/(2ae)`
`"or "(2)/(sqrt3)e=e^(2)-1`
`"or "sqrt3e^(2)-2e-sqrt3=0` ltBrgt `"or "e=(2pmsqrt(4+12))/(2sqrt3)=(2pm4)/(2sqrt3)`
`"or "e=(3)/(sqrt3)=sqrt3`
(c) Eccentricity of hyperbola`=(AB)/(PA-PB)=(6)/(4)=(3)/(2)`
If the eccentricity of conjugate hyperbola is e', then
`(1)/((3//2)^(2))+(1)/(e'^(2))=1`
`"or "e'=(3)/(sqrt5)`
(d)
Angle between the asymptotes is
`tan^(-1)|(2ab)/(a^(2)-b^(2))|=(pi)/(3)`
`"or "|(2xxa//b)/((a^(2)//b^(2)))|=sqrt3`
`"or "(2sqrt(e'^(2)-1))/(|e'^(2)-2|)=sqrt3`
where e' is the eccentricity of conjugate hyperbola. Therefore,
`e'=2.`