--- title: "Translate to (−1,0) and rotate by bisectors" subject: "Mathematics I" chapter: 3 key: "semester_1/mathematics_i:transform-axes-through-minus-1-0" source: https://www.bctengineer.com/q/semester_1/mathematics_i/transform-axes-through-minus-1-0 site: BCT Engineering --- # Translate to (−1,0) and rotate by bisectors Mathematics I · Chapter 3: Plane Analytic Geometry - Asked in: 2071 Shrawan, 5 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/mathematics_i/transform-axes-through-minus-1-0 - Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3 ## Question Transform the equation x² − 2xy + y² + x − 3y = 0 to axes through the point (−1, 0) parallel to the lines bisecting the angles between the original axes. ## Short answer Shift origin to (−1,0) then rotate 45°. Use x=−1+(X−Y)/√2, y=(X+Y)/√2. The equation reduces to 2Y²−√2 X=0, i.e., Y² = X/√2. ## Answer The new axes pass through the point (−1, 0) and are parallel to the lines bisecting the angles between the original axes, i.e. the lines y = x and y = −x. So the new origin is (−1, 0) and the new axes are rotated by 45°. #### Transformation formula For a shift of the origin to (h, k) = (−1, 0) followed by a rotation through θ = 45°: x = −1 + (X − Y)/√2 y = 0 + (X + Y)/√2 #### Substitute The given equation is (x − y)² + x − 3y = 0 Now - x − y = −1 + (X − Y)/√2 − (X + Y)/√2 = −1 − √2 Y - (x − y)² = 1 + 2√2 Y + 2Y² - x − 3y = −1 + (X − Y)/√2 − 3(X + Y)/√2 = −1 − √2 X − 2√2 Y Adding: 1 + 2√2 Y + 2Y² − 1 − √2 X − 2√2 Y = 2Y² − √2 X **The transformed equation is 2Y² − √2 X = 0, i.e. Y² = X/√2.** This is a parabola with its vertex at the new origin (−1, 0) and axis along the new X-axis (the line y = x + 1, through (−1, 0) with slope 1), latus rectum 1/√2. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which new axes does the question describe? A. Origin at (0,0), rotated 45° B. Origin at (−1,0), no rotation C. Origin at (1,0), rotated 90° D. Origin at (−1,0), rotated 45° Answer: D. Bisectors of the original axes are y = ±x. (Key point: New axes) 2. Shift and rotation formulae? A. x = 1 + (X−Y)/sqrt2, y = (X+Y)/sqrt2 B. x = (X−Y)/sqrt2, y = −1 + (X+Y)/sqrt2 C. x = −1 + (X−Y)/sqrt2, y = (X+Y)/sqrt2 D. x = −1 + X, y = Y Answer: C. Add the new origin to the rotated coordinates. (Key point: Formula) 3. Why write the equation as (x − y)^2 + x − 3y = 0? A. It removes the linear terms B. x − y becomes −1 − sqrt2 Y, avoiding X^2 C. It makes it a circle D. It is required by the formula Answer: B. x − y = −1 − sqrt2 Y. (Key point: Trick) 4. The transformed equation? A. 2Y^2 − sqrt2 X = 0 B. 2X^2 − sqrt2 Y = 0 C. 2Y^2 + sqrt2 X = 0 D. X^2 + Y^2 = 0 Answer: A. The parabola Y^2 = X/sqrt2 with vertex at the new origin. (Key point: Result) ## Related in Mathematics I - [Rotate axes 45° in 3x²+3y²+2xy=2](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Arotate-axes-3x2-3y2-2xy-45) — Mathematics I, chapter 3, asked 5× - [Axes along bisectors of 7x²+4xy+4y²=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Atransform-pair-of-lines-7x2-4xy-4y2) — Mathematics I, chapter 3 - [Rotate axes 30° in x²+2√3xy−y²=2a²](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Arotate-axes-x2-2sqrt3xy-y2-30) — Mathematics I, chapter 3, asked 5× - [Rotate axes by tan⁻¹2 in 4xy−3x²−a²=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Arotate-axes-4xy-3x2-tan-inverse-2) — Mathematics I, chapter 3, asked 5× - [Centre, axes of ellipse 2x²+3y²−4x−12y+13=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aellipse-2x2-3y2-centre-axes) — Mathematics I, chapter 3, asked 7× - [Centre, axes, directrices of 3x²+8xy−3y²−40x−20y+50=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aconic-3x2-8xy-3y2-directrices) — Mathematics I, chapter 3, asked 7× ## Related in other subjects - [Planes through line parallel to axes](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aplanes-through-line-parallel-axes) — Mathematics, chapter 3, asked 2× - [Lines intersect: point and plane through them](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Alines-intersect-point-plane-through) — Mathematics, chapter 3 - [Evaluate using Gauss' divergence theorem F=(2xy+z)i+y²j−(x+3y)k](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-gauss-divergence-theorem-f-four) — Mathematics III, chapter 2, asked 9× - [Evaluate using Gauss' divergence theorem F=(2xy+z)i+y²j−(x+3y)k](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-gauss-divergence-theorem-f-nine) — Mathematics III, chapter 2, asked 2×