Question

A point $\left( a,b \right)$ is called a good point if both a and b are integers. Number of good points on curve $xy=225$ are
(a) 20
(b) 18
(c) 16
(d) 14

Answer 1

Hint: First, we have to find factors of 225 which will be 1, 3, 5, 9, 15. Then, we have to make pairs like \[\left( 1,225 \right)\] and so on. Similarly, we have to interchange the position of point a and b and also count pairs like \[\left( 225,1 \right)\] and so on. Then, we have to also count negative integers because we know that \[-a\times -b=ab\] which results in a positive answer. Thus, at last we have to count all the pairs which will be our answer.

Complete step-by-step answer:
Here, we have to find the total number of points $\left( a,b \right)$ such that when multiplied results are equal to 225. So, first we will factor 225 i.e. given below.
$225=1\times 225$
$225=3\times 75$
$225=5\times 45$
$225=9\times 25$
$225=15\times 15$
So, here we get pairs which on multiplying results into 225 i.e. $\left( 1,225 \right),\left( 3,75 \right),\left( 5,45 \right),\left( 9,25 \right),\left( 15,15 \right)$
Now, we have to consider pairs by interchanging the position of b and a. By doing this we will get 4 more pairs i.e. $\left( 225,1 \right),\left( 75,3 \right),\left( 45,5 \right),\left( 25,9 \right)$ . Interchanging the $\left( 15,15 \right)$ will remain the same so we will count it as one pair only.
Till now we have a total of 9 pairs with us.
As, we have to multiply the point a and b, we should not forget negative numbers as we know that \[-a\times -b=ab\] . So, considering all the negative pairs, we will get again 9 more pairs given as:
\[\begin{align}
  & \left( -1,-225 \right),\left( -3,-75 \right),\left( -5,-45 \right),\left( -9,-25 \right),\left( -15,-15 \right), \\
 & \left( -225,-1 \right),\left( -75,-3 \right),\left( -45,-5 \right),\left( -25,-9 \right) \\
\end{align}\]
Thus, we have a total 18 pairs with us.
Number of good points on the curve $xy=225$ are 18.
Hence, option (b) is the correct answer.

Note: Remember that here we have nothing to do with the given curve $xy=225$ . So, do not plot a graph of this curve. Just we have to find factors of 225 and count the total number of pairs. Sometimes students forget to count negative integers and seeing integer words in question, only count positive points i.e. only 9 pairs will be there. So, do not make this mistake.