Question

If \[{x^2} - hx - 21 = 0,{x^2} - 3hx + 35 = 0(h > 0)\] has a common root, then the value of \[h\] is equal to
A. 1
B. 2
C. 3
D. 4

Answer 1

Hint:
In our case, there is an equation in the question. The ratio of the variables in the generalized equation must be determined. They share a common root, which is the condition. Therefore, we must identify the discriminant in order to determine if the roots are made up or real. The ratio for the variables can then be determined by locating the common root.

Complete Step-By-Step Solution:
We are given two equations in the question
\[{x^2} - hx - 21 = 0\]------ (1)
\[{x^2} - 3hx + 35 = 0\]-------- (2)
And we have been given the condition that,
\[(h > 0)\]
Here, we have to determine the value of \[h\]
Now, let us subtract the given two equations, we have
\[\left( {{x^2} - hx - 21} \right) - \left( {{x^2} - 3hx + 35} \right) = 0\]
On subtracting the above two equations, we get
\[2hx = 56\]
Now, we have to calculate the value for \[hx\] we get
\[hx = 28\]
Now, we have to substitute the value of \[hx\] in first equation, we get
\[{x^2} - 28 - 21 = 0\]
Now, let us simplify the like terms, we get
\[{x^2} = 49\]
On taking square on both sides of the above equation, we get
\[x = \pm 7\]
It is already known that, we take only positive values for \[x\]
Since, the condition is
\[h > 0\]
Now, we have to substitute the value of \[x = 7\] in \[hx = 28\] we get
\[h(7) = 28\]
On solving for \[h\] we get
\[h = \frac{{28}}{7}\]
Now, on further simplification we obtain
\[h = 4\]
Therefore, If \[{x^2} - hx - 21 = 0,{x^2} - 3hx + 35 = 0(h > 0)\] has a common root, then the value of \[h\] is equal to \[h = 4\]
Hence, the option D is correct

Note:
Students often tend to make mistake in these types of problems, because here it is asked to determine the common root. Students should keep in mind that finding the discriminant is not specifically asked for in the question. The crucial phrase is "common root." We don't even have to look for the actual origins. Simply comparing the types of roots in two equations will do.