The triangle PQR of area ‘A’ is inscribed in the parabola y2 = 4ax such that the vertex P lies at the vertex of the parabola and the base QR is a focal chord. The modulus of the difference of the ordinates of the points Q and R is :
Question
The triangle PQR of area ‘A’ is inscribed in the parabola y2 = 4ax such that the vertex P lies at the vertex of the parabola and the base QR is a focal chord. The modulus of the difference of the ordinates of the points Q and R is :
Solution
Let's solve the problem step by step:
Step 1: Understand the given information. We are given a triangle PQR that is inscribed in the parabola y^2 = 4ax. The vertex P of the triangle lies at the vertex of the parabola, and the base QR is a focal chord.
Step 2: Find the coordinates of the vertex P. Since the vertex P lies at the vertex of the parabola, its coordinates can be found by substituting x = 0 into the equation of the parabola. When x = 0, we have y^2 = 4a(0), which simplifies to y^2 = 0. This implies that y = 0. Therefore, the coordinates of the vertex P are (0, 0).
Step 3: Find the equation of the line QR. Since QR is a focal chord, it passes through the focus of the parabola. The focus of the parabola y^2 = 4ax is located at the point (a/4, 0). Using the two points P(0, 0) and the focus (a/4, 0), we can find the equation of the line QR using the point-slope form. The slope of the line QR can be found as (0 - 0) / (a/4 - 0) = 0. Therefore, the equation of the line QR is y = 0.
Step 4: Find the coordinates of points Q and R. Since the equation of the line QR is y = 0, the y-coordinate of both points Q and R is 0. To find the x-coordinate of points Q and R, we substitute y = 0 into the equation of the parabola y^2 = 4ax. This gives us 0^2 = 4ax, which simplifies to 0 = 4ax. Since a is a non-zero constant, we can conclude that x = 0. Therefore, the coordinates of both points Q and R are (0, 0).
Step 5: Find the difference of the ordinates of points Q and R. The ordinate of a point is its y-coordinate. Since both points Q and R have a y-coordinate of 0, the difference of their ordinates is 0 - 0 = 0.
Step 6: Find the modulus of the difference of the ordinates. The modulus of a number is its absolute value. Since the difference of the ordinates is 0, the modulus of the difference is also 0.
Therefore, the modulus of the difference of the ordinates of points Q and R is 0.
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