Question

If [1,p/3] is the mid point of the line segment joining the points [2,0)

And [1,2/9] then show that the line 5x+3y+2=0 passes through the point (-1,3p)

If 2 positive integers p and q are written as p=a cube b cube

And q =a cube b . A and b are prime numbers then verify lcm (p,q)

*HCF (p,q)=pq

Answer 1

Midpoint of the line segment joining points (2,0) and (0,2/9) will be
[(2+0)/2 , (0+2/9)/2] = (1,1/9)
As it is given that 1,p/3 is the mid point. so
p/3 = 1/9
i.e. p=1/3
Now we have to show that ( -1,3p) = (-1,1) passes through 5x+3y+2=0. so putting the value of x and y as -1 and 1 respectively in the equation, Taking Left Hand side:

= 5*(-1)+3(1)+2
= -5+3+2
= -5+5
= 0 that is equal to the Right hand side

HCF of two or more numbers is the product of the smallest power of each common prime factors involved in the numbers.

LCM of two or more numbers is a product of the greatest power of its prime factors involved in the numbers with highest power.


SOLUTION:

Given:

p = a²b³

q= a³b

HCF(p,q)= a²b

LCM (p,q)= a³b³

HCF× LCM= a²b× a³b³= a^5b⁴

HCF× LCM=a^5b⁴...........(1)

p ×q= a²b³× a³b= a^5b⁴

p ×q= = a^5b⁴..................(2)

Lcm (p,q) × Hcf(p,q) = pq

a^5b⁴ = a^5b⁴
[From equation 1 and 2]

Verified,..