--- title: "Condition for x cosα + y sinα = p to touch ellipse" subject: "Mathematics I" chapter: 3 key: "semester_1/mathematics_i:ellipse-tangent-condition-xcos-ysin" source: https://www.bctengineer.com/q/semester_1/mathematics_i/ellipse-tangent-condition-xcos-ysin site: BCT Engineering --- # Condition for x cosα + y sinα = p to touch ellipse Mathematics I · Chapter 3: Plane Analytic Geometry - Asked in: 2070 Asar - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/mathematics_i/ellipse-tangent-condition-xcos-ysin - Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3 ## Question Find the condition that the straight line x cos α + y sin α = p touches the ellipse x²/a² + y²/b² = 1. ## Short answer The line x cosα + y sinα = p touches the ellipse x²/a² + y²/b² = 1 if p² = a² cos²α + b² sin²α. ## Answer The line x cosα + y sinα = p is in the normal (perpendicular) form, with p the distance of the line from the origin. #### Using the tangent condition Write it as x cosα + y sinα − p = 0, so l = cosα, m = sinα, n = −p. The line lx + my + n = 0 touches x²/a² + y²/b² = 1 if n² = a²l² + b²m² (shown below), hence **p² = a² cos²α + b² sin²α** #### Proof of the condition Write the line as y = (p − x cosα)/sinα and put in b²x² + a²y² = a²b²: b²x² sin²α + a²(p − x cosα)² = a²b² sin²α (b² sin²α + a² cos²α)x² − 2a²p cosα · x + a²(p² − b² sin²α) = 0 A tangent gives equal roots (discriminant = 0): 4a⁴p² cos²α = 4(b² sin²α + a² cos²α)·a²(p² − b² sin²α) a²p² cos²α = (b² sin²α + a² cos²α)p² − b² sin²α (b² sin²α + a² cos²α) 0 = b² sin²α · p² − b² sin²α (b² sin²α + a² cos²α) p² = a² cos²α + b² sin²α Hence the line x cosα + y sinα = p touches the ellipse iff **p² = a²cos²α + b²sin²α**. (Check: for α = 0 the line x = p touches the ellipse at x = ±a, giving p² = a² ✓.) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In x cosα + y sinα = p, what is p? A. The perpendicular distance of the line from the origin B. The slope C. The intercept on x-axis D. The angle Answer: A. It is the normal form. (Key point: Identification) 2. Compared with lx + my + n = 0, what are l, m, n? A. sinα, cosα, p B. cosα, sinα, p C. 1, 1, −p D. cosα, sinα, −p Answer: D. Rewrite as x cosα + y sinα − p = 0. (Key point: Mapping) 3. Condition to touch the ellipse? A. p^2 = a^2 sin^2α + b^2 cos^2α B. p = a cosα + b sinα C. p^2 = a^2 cos^2α + b^2 sin^2α D. p^2 = a^2 cos^2α − b^2 sin^2α Answer: C. From n^2 = a^2l^2 + b^2m^2. (Key point: Result) 4. Why does α = 0 check the result? A. x = p touches at b B. x = p touches at x = ±a, so p^2 = a^2 C. It gives p = 0 D. It gives a circle Answer: B. p^2 = a^2 when cosα = 1. (Key point: Check) ## Related in Mathematics I - [Condition for lx+my+n=0 to be normal to ellipse](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aellipse-normal-condition-lx-my-n) — Mathematics I, chapter 3, asked 2× - [Condition for lx+my+n=0 to be tangent to ellipse](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aellipse-tangent-condition-lx-my-n) — Mathematics I, chapter 3 - [Beta-Gamma: ∫ cos⁴3θ sin²6θ dθ = 5π/192](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Abeta-gamma-cos4-3theta-sin2-6theta) — Mathematics I, chapter 2, asked 3× - [Gamma function: ∫ sin⁶(x/2) cos⁶(x/2) dx](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Agamma-sin6-cos6-half-x) — Mathematics I, chapter 2, asked 3× - [DUIS: ∫ dx/(a²sin²x+b²cos²x)²](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aduis-dx-over-a2sin2-b2cos2-squared) — Mathematics I, chapter 2, asked 2× - [Beta-Gamma: ∫ sin⁴x cos²x dx = (3π−4)/192](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Abeta-gamma-sin4-cos2-pi4) — Mathematics I, chapter 2, asked 2× ## Related in other subjects - [Second region condition, ellipse midpoint](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aellipse-second-region-condition) — Computer Graphics, chapter 2 - [RK-2: y' = sin x + cos y](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Ark2-sinx-cosy) — Numerical Method, chapter 6 - [Fit straight line: humidity and moisture](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Ahumidity-moisture-straight-line-fit) — Probability and Statistics, chapter 5 - [DDB with switchover to straight line](https://www.bctengineer.com/question.md?key=semester_6%2Fengineering_economics%3Addb-switchover-30000-7yr) — Engineering Economics, chapter 7