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If the roots of the equation kx2 – 3x -1= 0 are the reciprocal of the roots of the equation x2 + 3x – 4 = 0 then K =
- a)4
- b)-4
- c)3
- d)-3
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
If the roots of the equation kx2– 3x -1= 0 are the reciprocal of...
Understanding the Problem
To solve for k in the given equations, we need to analyze the relationships between the roots of the two quadratic equations provided.
Step 1: Find Roots of the Second Equation
The second equation is:
x^2 + 3x - 4 = 0
We can use the quadratic formula:
Roots = [-b ± sqrt(b^2 - 4ac)] / 2a
Here, a = 1, b = 3, c = -4.
Calculating the discriminant:
- Discriminant = b^2 - 4ac = 3^2 - 4(1)(-4) = 9 + 16 = 25
Now, finding the roots:
- Roots = [-3 ± sqrt(25)] / 2(1) = [-3 ± 5] / 2
Calculating the roots:
- Root 1 = (2)/2 = 1
- Root 2 = (-8)/2 = -4
Thus, the roots are 1 and -4.
Step 2: Find the Reciprocals
The reciprocals of these roots are:
- Reciprocal of 1 = 1
- Reciprocal of -4 = -1/4
Step 3: Set Up the First Equation
The first equation is:
kx^2 - 3x - 1 = 0
Given that the roots of this equation are the reciprocals (1 and -1/4), we can derive relationships using Vieta's formulas:
- Sum of roots = 1 + (-1/4) = 3/4
- Product of roots = 1 * (-1/4) = -1/4
Using Vieta's formulas for the first equation:
- Sum of roots = 3/k
- Product of roots = -1/k
Step 4: Set Up Equations
From the sum of roots:
3/k = 3/4
=> k = 4
From the product of roots:
-1/k = -1/4
=> k = 4
Conclusion
Both calculations yield k = 4, confirming that the correct answer is option 'A'.
To solve for k in the given equations, we need to analyze the relationships between the roots of the two quadratic equations provided.
Step 1: Find Roots of the Second Equation
The second equation is:
x^2 + 3x - 4 = 0
We can use the quadratic formula:
Roots = [-b ± sqrt(b^2 - 4ac)] / 2a
Here, a = 1, b = 3, c = -4.
Calculating the discriminant:
- Discriminant = b^2 - 4ac = 3^2 - 4(1)(-4) = 9 + 16 = 25
Now, finding the roots:
- Roots = [-3 ± sqrt(25)] / 2(1) = [-3 ± 5] / 2
Calculating the roots:
- Root 1 = (2)/2 = 1
- Root 2 = (-8)/2 = -4
Thus, the roots are 1 and -4.
Step 2: Find the Reciprocals
The reciprocals of these roots are:
- Reciprocal of 1 = 1
- Reciprocal of -4 = -1/4
Step 3: Set Up the First Equation
The first equation is:
kx^2 - 3x - 1 = 0
Given that the roots of this equation are the reciprocals (1 and -1/4), we can derive relationships using Vieta's formulas:
- Sum of roots = 1 + (-1/4) = 3/4
- Product of roots = 1 * (-1/4) = -1/4
Using Vieta's formulas for the first equation:
- Sum of roots = 3/k
- Product of roots = -1/k
Step 4: Set Up Equations
From the sum of roots:
3/k = 3/4
=> k = 4
From the product of roots:
-1/k = -1/4
=> k = 4
Conclusion
Both calculations yield k = 4, confirming that the correct answer is option 'A'.
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Community Answer
If the roots of the equation kx2– 3x -1= 0 are the reciprocal of...
∵ x2 + 3x – 4 = 0
or; x2 – 4x + x – 4 = 0
or; x(x – 4) + 1(x – 4) = 0
or; (x – 4)(x + 1) = 0
x = 4; -1
Eqn. having roots 1/2 & 1/−1 = 1/4 & – 1 is.
or x2 – (1/4 – 1) + 1/4(-1) = 0
or x2 + 3/4x – 1/4 = 0
Multiplying by 4 ; we get
4x2 + 3x -1 = 0
Comparing it with kx2 + 3x -1 = 0
We get K = 4
Tricks : Eqn. having roots the reciprocal of the roots of ax2 + bx + c = 0 is cx2 + bx +a = 0 i.e. 1st and last term interchanges.
or; x2 – 4x + x – 4 = 0
or; x(x – 4) + 1(x – 4) = 0
or; (x – 4)(x + 1) = 0
x = 4; -1
Eqn. having roots 1/2 & 1/−1 = 1/4 & – 1 is.
or x2 – (1/4 – 1) + 1/4(-1) = 0
or x2 + 3/4x – 1/4 = 0
Multiplying by 4 ; we get
4x2 + 3x -1 = 0
Comparing it with kx2 + 3x -1 = 0
We get K = 4
Tricks : Eqn. having roots the reciprocal of the roots of ax2 + bx + c = 0 is cx2 + bx +a = 0 i.e. 1st and last term interchanges.
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