JEEOlympiad
The conic 6x^2 - 4xy + 9y^2 - 20x + 20y - 20 = 0 is an ellipse. If…
Question
The conic 6x^2 - 4xy + 9y^2 - 20x + 20y - 20 = 0 is an ellipse. If the square of its eccentricity equals m/n in lowest terms, find m+n.
✓ Verified answer: 3checked by our engine — not a guess
Step-by-step solution
Step 1 — translate to the centre. Set the partial derivatives to zero:
F_x = 12x - 4y - 20 = 0, F_y = -4x + 18y + 20 = 0.
Solving: from the first 3x - y = 5; from the second -2x + 9y = -10.
Then y = 3x - 5, substitute: -2x + 9(3x-5) = -10 => 25x = 35 => x = 7/5, y = 3(7/5)-5 = -4/5.
Centre = (7/5, -4/5).
The constant after translation is F(centre).
Evaluating F at the centre gives -42, so in centred coordinates X^T Q X = 42, where Q = [[6,-2],[-2,9]] is the matrix of the quadratic part (off-diagonal entries are half the xy-coefficient).
Step 2 — diagonalise Q. Eigenvalues solve (6-λ)(9-λ) - 4 = 0 => λ^2 - 15λ + 50 = 0 => λ = 5 or λ = 10.
The ellipse becomes 5 X'^2 + 10 Y'^2 = 42, i.e. X'^2/(42/5) + Y'^2/(42/10) = 1.
So the squared semi-axes are 42/5 (major) and 42/10 = 21/5 (minor): a^2 = 42/5, b^2 = 21/5.
Step 3 — eccentricity. e^2 = 1 - b^2/a^2 = 1 - (21/5)/(42/5) = 1 - 1/2 = 1/2.
m/n = 1/2, m+n = 3.
Final answer3
Stuck on a problem like this?
Paste any JEE or NEET question — verified working, a confidence %, and an honest “not sure” instead of a bluff.
Solve my doubt →More Coordinate Geometry solutions
A ray of light starts at the point A(2,3). It first strikes the…JEE · MathA point P moves in the plane so that the length of the tangent drawn…JEE · MathIn triangle ABC the vertices are A(0,0), B(14,0), C(5,12). Consider…JEE · MathConsider the ellipse x^2/9 + y^2/4 = 1 with foci S and S'. Let P be…JEE · MathA circle passes through the two points (1,0) and (5,0) and is tangent…JEE · MathPQ is a focal chord of the parabola y^2 = 8x with focus S. The length…JEE · MathA single straight line is a common tangent to both parabolas y^2 = 4x…JEE · MathFor the hyperbola x^2/16 - y^2/9 = 1, the tangent at a variable point…JEE · Math