Question
Question: If the line \(y=\sqrt{3}x\) cuts the curve \({{x}^{3}}+a{{x}^{2}}+bx-72=0\) at \(A,B\) and \(C\), th...
If the line y=3x cuts the curve x3+ax2+bx−72=0 at A,B and C, the OA×OB×OC (where O is origin) is:
A. 576
B. −576
C. a+b−c−576
D. a+b+c−576
Solution
To solve this question, we will find the value of x from the equation of line y=3x that will be x=3y. Then, we will substitute the value of x in the equation of curve x3+ax2+bx−72=0 and will find the relation between its roots that will be also the relation between coordinates of A,B and C. Then, we will use OA×OB×OC and substitute the corresponding values to simplify it and will find the required answer.
Complete step-by-step solution:
Since, given that the equation of line:
⇒y=3x
After simplifying it, we will get the value of x as:
x=3y
Now, we will use x=3y in the equation of curve as:
⇒(3y)3+a(3y)2+b(3y)−72=0
Here, we will simplify the above expression as:
⇒33y3+a3y2+b3y−72=0
Now, we will multiply with 33 in the above equation and will simplify it as: