If the line 3x−2y+12=0 intersects the parabola 4y=3x2 at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to
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Step-by-Step Solution
Step 1: Find the intersection points A and B
First, we need to find the coordinates of the intersection points A and B. We can do this by substituting the expression for y from the linear equation into the parabolic equation. The line equation is 3x−2y+12=0, which can be rewritten as y=23x+12. Substitute this into the parabola equation 4y=3x2.
Step 2: Solve the quadratic equation for x
Simplify the equation to get a quadratic equation in terms of x. After simplifying, we get 3x2−6x−24=0, which can be further simplified to x2−2x−8=0. Factoring this quadratic equation gives us (x−4)(x+2)=0, yielding x -values of 4 and −2.
Step 3: Find the coordinates of A and B
Now, substitute these x -values back into the linear equation y=23x+12 to find the corresponding y -values. For x=4, y=12, so point A is (4,12). For x=−2, y=3, so point B is (−2,3).
Step 4: Identify the vertex of the parabola
The equation of the parabola is 4y=3x2, which can be written as y=43x2. This is a parabola of the form y=ax2, which has its vertex at the origin (0,0). Let's call the vertex V(0,0).
Step 5: Calculate the slopes of VA and VB
To find the angle subtended by the line segment AB at the vertex V, we need to find the slopes of the lines VA and VB. The slope of VA is 4−012−0=3. The slope of VB is −2−03−0=−23.
Step 6: Calculate the angle using the tangent formula
The angle θ between two lines with slopes m1 and m2 is given by the formula tanθ=1+m1m2m1−m2. Substituting the slopes mVA=3 and mVB=−23, we get tanθ=79. Therefore, the angle θ is tan−1(79).