Question

Trace the conic 17x2 - 12xy + 8y2 + 46x - 28y +17 = 0. ​

Answer 1

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Answer 2

For 17x^2 - 12xy + 8y^2 + 46x - 28y +17 = 0, ​the conic is central.

Given,

17x2 - 12xy + 8y2 + 46x - 28y +17 = 0. ​

To find,

The conics of 17x2 - 12xy + 8y2 + 46x - 28y +17 = 0. ​

Solution,

• According to a theorem,

Let ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 be a central conic.

Then its center is the intersection of the lines

• Here, this theorem tells us that if ax + hy + g = 0 and hx + by + f = 0 intersect, then the conic is central and the point of intersection of these straight lines is the center of the conic.

• But in case the lines don't intersect, then the conic under consideration can't be central, that is, it is non-central.

• Hence, the conic is non-central if the slopes of these lines are equal, that is, if ab = h2.

• So, we have the result:

The conic ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 is

1. central if ab  ≠ h^2

2. non-central if ab = h^2

• Now, the given expression 17x^2 - 12xy + 8y^2 + 46x - 28y +17 = 0. ​

is in the form of ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0.

• From this we get,

a = 17, b = 8, h = -6, g = 23, f = -14

here,

ab ≠ h^2

17 * 8  ≠ (-6)^2

136 ≠ 36

So, ab  ≠ h^2 and the conic are central.

The intersection of the lines is its center,

17x - 6y +23 =0 -- equation (1)

3x - 4y + 7 = 0 -- equation (2)

Solving the above two equations using the elimination method,

Multiply  equation (1) with 4, and  equation (2) with 6, and subtract them, we get,

68x - 24y + 92 = 0

-

18x - 24y + 42 = 0

=

50x = 50

x = 1

Now, substitute x =1 in equation (2), we get.,

3 (1) - 4y + 7=0

-4y = 4

y= -1

So, the values of x=1, y= -1

and the conic of 17x^2 - 12xy + 8y^2 + 46x - 28y +17 = 0 is central.

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