Question:

Let ABC\triangle ABC be an isosceles triangle in which AA is at (1,0)(-1, 0), A=2π3\angle A = \frac{2\pi}{3}, AB=ACAB = AC, and BB is on the positive xx-axis. If BC=43BC = 4\sqrt{3} and the line BCBC intersects the line y=x+3y = x + 3 at (α,β)(\alpha, \beta), then β4α2\frac{\beta^4}{\alpha^2} is:

Updated On: Mar 20, 2025
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Correct Answer: 36

Solution and Explanation

Given:
A=(1,0),A=2π3,AB=AC,andBC=43 A = (-1, 0), \quad \angle A = \frac{2\pi}{3}, \quad AB = AC, \quad \text{and} \quad BC = 4\sqrt{3}

Step 1: Placing Points B B and C C
Since B B is on the positive x x -axis and ABC \triangle ABC is isosceles with AB=AC AB = AC , the coordinates of B B can be represented as:
B=(x,0),x>1 B = (x, 0), \quad x > -1
The angle A=2π3 \angle A = \frac{2\pi}{3} implies that the line AC AC makes an angle of 2π3 \frac{2\pi}{3} with the positive x x -axis. Thus, the slope of line AC AC is:
tan(2π3)=3 \tan\left(\frac{2\pi}{3}\right) = -\sqrt{3}
Let the coordinates of C C be (xc,yc) (x_c, y_c) . Since AB=AC AB = AC and BC=43 BC = 4\sqrt{3} , we can use the distance formula to find x x and the coordinates of C C .

Step 2: Calculating the Lengths
The length of AB AB is given by:
AB=x+1 AB = |x + 1|
Similarly, the length of AC AC is also x+1 |x + 1| .
Given that BC=43 BC = 4\sqrt{3} , we find the coordinates of C C such that it satisfies the isosceles condition and the length of BC BC .

Step 3: Equation of Line BC BC
The line BC BC can be represented in the form:
y=mx+c y = mx + c
where m m is the slope and c c is the intercept. Using the coordinates of B B and C C , we can find the equation of line BC BC .

Step 4: Intersection with Line y=x+3 y = x + 3
The line BC BC intersects the line y=x+3 y = x + 3 at (α,β) (\alpha, \beta) . Substituting the equation of BC BC into y=x+3 y = x + 3 and solving for α \alpha and β \beta gives the required values.

Step 5: Calculating β4α2 \frac{\beta^4}{\alpha^2}
After finding α \alpha and β \beta , we compute:
β4α2 \frac{\beta^4}{\alpha^2}
Given that the solution yields:
β4α2=36 \frac{\beta^4}{\alpha^2} = 36

Conclusion: The value of β4α2 \frac{\beta^4}{\alpha^2} is 36.

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