Attempt following question by selecting a choice to answer.
e
Find the transformed equation of
5x2+3xy+y2 = 5. Identify the type of conic that the transformed equation would represent.
D eA.
ex′2(1011) +
y′210 = 1, ellipse
B.
ex′2(1011) +
y′210 = 1, circle
C.
ex′2(1011) +
y′210 = 1, parabola
D.
eNone of the above
e
A
Step 1: To know the conic represented by x2+3xy+y2 = 5, we find the transformed equation of it by removing x′(y)′ - term.
Step 2: A = 5, B = 3, and C = 1[Compare with the standard second degree equation.]
Step 3: The angle to which the axes be rotated to remove the x′ (y)′ - term in the transformed equation of Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is θ = 12 Cot-1[A-CB].
Step 4: = 12 Cot-1[43] ⇒ cot 2θ = 43[Substitute the values of A, B, and C.]
Step 5: tan 2θ = 34 ⇒ 2tanθ1-tan2θ = 34 ⇒ 3tan2 θ + 8tan θ - 3 = 0 ⇒ tan θ = 13[As θ is an acute angle, reject tan θ = - 3.]
Step 6: Then cos θ = 310, sin θ = 110[Find cos θ , sin θ from tan θ.]
Step 7: When the axes are turned through an angle θ such that tan θ = 13, then x = x′ cos θ - y′ sin θ = 3x′ - y′10, y = x′ sin θ + y′ cos θ = x′+3y′10.[Substitute the values of sin θ, cos θ.]
Step 8: The transformed equation of the conic is 5[3x′-y′10]² + 3[3x′-y′10][x′+3y′10] + [x′+3y′10]² = 5
Step 9: 55(x′)2 + 5(y′)2 = 50
Step 10: x′2(1011) + y′210 = 1, which represents an ellipse.[Divide both sides by 50.]