Question

If the length of the latus rectum of a parabola whose focus is (a,a) and tangent at its vertex is x + y = a, is 16. Then |a| is equal to:

(a) 2√3

(b) 2√2

(c) 4√2

(d) 4

Answer 1

To find the value of |a|, we need to determine the length of the latus rectum of the parabola. The length of the latus rectum is 16, which can be obtained by substituting the equation of the tangent line into the equation of the parabola. Solving for |a|, we find that |a| = 0.

Step-by-step explanation:

To find the value of |a|, we need to determine the length of the latus rectum of the parabola. The latus rectum is a line segment perpendicular to the axis of symmetry of the parabola and passes through the focus. In this case, the focus is located at (a, a). Since the tangent at the vertex (0, 0) is given by x + y = a, we can substitute y = a - x into the equation of the parabola y = ax + bx^2 to find the points of intersection. Solving the quadratic equation, we get a unique solution for x = 16. This means that the length of the latus rectum is 16.

The equation of a parabola with a known length of the latus rectum is given by y = ±(4a)x. Substituting the length of the latus rectum, we have 16 = ±(4a)(0). From this equation, we can determine the value of a by solving for a.

16 = ±(4a)(0)

16 = ±0

Since 0 cannot equal 16, we conclude that there is no solution for a. Therefore, |a| = 0.