st","extracted_subject":"MATH","extracted_chapter":"CONIC SECTIONS","extracted_topic":"PARABOLA","questionPreviewUrl":"https://cdn.pureessence.tech/question-previews/packed_canvases/canvas_1816.png?top_left_x=532&top_left_y=0&width=532&height=264&app=solveeit"},"seo_metadata":{"keywords":[]},"doubt_solver_data":{"concept_summary":[]},"embeddings":[],"is_hidden":false,"_id":"66d23e71425d76b7ab31db7f","slug":"the-normals-at-left-x_1y_1-rightleft-x_2y_2-right-","content":"The normals at \\(\\left( x_{1},y_{1} \\right),\\left( x_{2},y_{2} \\right)\\) on the parabola\n\n\\(y^{2} = 12x\\), meet again on the parabola, then x₁x₂ + y₁y₂=","options":[{"_id":"68a3a0b47f15af4716bc642d","text":"147","sequence":0,"isCorrect":false},{"_id":"68a3a0b47f15af4716bc642e","text":"108","sequence":1,"isCorrect":true},{"_id":"68a3a0b47f15af4716bc642f","text":"27","sequence":2,"isCorrect":false},{"_id":"68a3a0b47f15af4716bc6430","text":"–27","sequence":3,"isCorrect":false}],"correct_answer":"108","explanation":"If the normals at t₁ and t₂ intersect on the parabola, then t₁t₂ = 2.\n\n∴ x₁x₂ + y₁y₂ = \\(at_{1}^{2}.at_{2}^{2} + 2at_{1}.2at_{2}\\)\n\n= a²(t₁.t₂)² + 4a²(t₁.t₂)\n\n= 12a² = 108","question_type":"single_choice","appeared_in":[],"created_at":"2024-08-30T21:49:37.021Z","__v":0,"vector_store_id":"5d776585-f553-4e36-9be6-88336ee2bf6a","updated_at":"2024-08-30T21:49:37.021Z","isStarred":false},"params":{"questionSlug":"the-normals-at-left-x_1y_1-rightleft-x_2y_2-right-"}}]}]]

Question: The normals at \(\left( x_{1},y_{1} \right),\left( x_{2},y_{2} \right)\) on the parabola \(y^{2} = ...

Solution