# A parabola with focus $(-1,-1)$ is tangent to $y=3x-8$ at $(7,13)$. Find the latus rectum. [duplicate]

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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.

## marked as duplicate by amd, HK Lee, Claude Leibovici, Chris Custer, Martin SleziakMay 11 '18 at 11:20

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• I tried the parabola of the form $x^2=4ay$ considering that (7,13) lies on parabola and focus us (-1,-1) – Samar Imam Zaidi May 11 '18 at 4:34
• But not getting the equation of parabola – Samar Imam Zaidi May 11 '18 at 4:35
• Let $(p,q)$ be the vertex, what is the equation of directrix & that of the parabola – lab bhattacharjee May 11 '18 at 4:44
• Why do you think the parabola’s axis is vertical? – amd May 11 '18 at 6:29

## 1 Answer

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.