On comparing equation the above two equations, we have
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Now, the equation of the tangent to the parabola at point is given by
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So, the equation of tangents at and are
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Since tangent at the vertex is axis whose equation is
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to find put in
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[ otherwise tangent at will be ]
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Similarly
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Given that
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Let, be any point on the required locus
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Then is the point of intersection of tangents at and
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To find we solve and
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Subtracting from we have
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[ and are distinct ]
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i.e.,
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from , we get
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We know that
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[Using ]
\n\n\n\n\n\n\n\n
Hence, the equation of the locus of the point of intersection of two tangents is .
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\n"},"comment":{"@type":"Comment","text":"
We need to find the equations of tangents drawn to the parabola at points and in the parametric form. And, then solve those two equations to find the point of intersection, but it will be containing the parameters, which need to be eliminated using the given condition, that the distance and also, using the identity,
\n\n\n"},"encodingFormat":"text/markdown","learningResourceType":"Practice problem","suggestedAnswer":[],"text":"Two tangents to the parabola meet the tangents at the vertex in the point and If prove that the equation of the locus of the point of intersection of two tangent is "},"name":"Quiz on Conic Sections","typicalAgeRange":"10-17","url":"https://www.embibe.com/questions/Two-tangents-to-the-parabola-y2%C2%A0%3D%C2%A08x-meet-the-tangents-at-the-vertex-in-the-point-P-and-Q.-If-PQ%3D4%2C-prove-that-the-equation-of-the-locus-of-the-point-of-intersection-of-two-tangent-is-y2%C2%A0%3D%C2%A08%28x%C2%A0%2B%C2%A02%29.%C2%A0/EM6785903"}
Maharashtra Board Mathematics Solutions for EXERCISE 7.1
Simple step-by-step solutions to EXERCISE 7.1 questions of Conic Sections from Mathematics and Statistics (Arts & Science) Part 1. Also get 3D topic explainers, cheat sheets, and unlimited doubts solving on EMBIBE.
Questions from EXERCISE 7.1 with Hints & Solutions
Two tangents to the parabola meet the tangents at the vertex in the point and If prove that the equation of the locus of the point of intersection of two tangent is
The tower of a bridge, hung in the form of a parabola have their tops meters above the roadway and are meters apart. If the cable is meters above the road way at the centre of the bridge, find the length of the vertical supporting cable from the centre.