Determine the focus coordinates, the axis of the parabola, the equation of the directrix and the latus rectum length for y2 = -8x

As an experienced tutor registered on UrbanPro, I'd be happy to assist you with this question. UrbanPro is a fantastic platform for finding online coaching tuition, offering a wide range of subjects and experienced tutors. Now, let's delve into your question regarding the parabola with the equation y2=−8xy2=−8x. To determine the focus coordinates, axis of the parabola, equation of the directrix, and the length of the latus rectum, we'll use some fundamental concepts of conic sections. Focus Coordinates: For the given equation y2=−8xy2=−8x, we can see that it's a parabola opening towards the left with its focus to the left of the vertex. The focus coordinates are (p,0)(p,0) where pp is the distance from the vertex to the focus. Since the coefficient of xx is negative, we have p=−14p=−41, so the focus coordinates are (−1/4,0)(−1/4,0). Axis of the Parabola: The axis of the parabola is a straight line passing through the vertex and the focus. Since the parabola opens to the left, the axis is a vertical line given by the equation x=−px=−p. Therefore, the axis of the parabola is x=1/4x=1/4. Equation of the Directrix: The directrix is a vertical line perpendicular to the axis of the parabola and equidistant from the vertex but in the opposite direction from the focus. Since the parabola opens to the left, the directrix is a vertical line given by the equation x=−px=−p. Therefore, the equation of the directrix is x=−1/4x=−1/4. Length of the Latus Rectum: The latus rectum is a line segment perpendicular to the axis of the parabola through the focus and whose endpoints lie on the parabola. Its length is equal to the absolute value of the coefficient of xx in the equation of the parabola. Therefore, the length of the latus rectum is ∣4p∣=1∣4p∣=1. So, summarizing: Focus Coordinates: (−1/4,0)(−1/4,0) Axis of the Parabola: x=1/4x=1/4 Equation of the Directrix: x=−1/4x=−1/4 Length of the Latus Rectum: 11 These parameters provide key insights into the geometry and behavior of the given parabola. If you have any further questions or need clarification, feel free to ask! read less