Let $y=3x-8$ be the equation of tangent at the point $(7,13)$ lying on the parabola whose focus is $(-1,-1)$. Find the length of the latus rectum of the parabola.
2026-03-25 17:41:49.1774460509
A parabola with focus $(-1,-1)$ is tangent to $y=3x-8$ at $(7,13)$. Find the latus rectum.
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A geometric approach: The length of the latus rectum is four times the distance from the focus to the vertex. Drop a perpendicular from the focus to the tangent line. The foot of this perpendicular lies on the tangent through the parabola’s vertex. Use the reflective property of parabolas to find the direction of its axis (neither horizontal nor vertical in this case); the tangent through the vertex is perpendicular to this.