
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
164.1k+ views
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.
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.
Recently Updated Pages
Trigonometry Formulas: Complete List, Table, and Quick Revision

Difference Between Distance and Displacement: JEE Main 2024

IIT Full Form

Uniform Acceleration - Definition, Equation, Examples, and FAQs

Difference Between Metals and Non-Metals: JEE Main 2024

Newton’s Laws of Motion – Definition, Principles, and Examples

Trending doubts
JEE Main Marks Vs Percentile Vs Rank 2025: Calculate Percentile Using Marks

JEE Mains 2025 Cutoff: Expected and Category-Wise Qualifying Marks for NITs, IIITs, and GFTIs

NIT Cutoff Percentile for 2025

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025 CutOff for NIT - Predicted Ranks and Scores

Other Pages
NCERT Solutions for Class 10 Maths Chapter 13 Statistics

NCERT Solutions for Class 10 Maths Chapter 11 Areas Related To Circles

NCERT Solutions for Class 10 Maths Chapter 12 Surface Area and Volume

NCERT Solutions for Class 10 Maths Chapter 14 Probability

NCERT Solutions for Class 10 Maths In Hindi Chapter 15 Probability

Total MBBS Seats in India 2025: Government and Private Medical Colleges
