What is the equation for this parabola?

Question

It includes the points (10,0), (-2,0), & (12,4)
and the axis of symmetry line has a slope = 1.

Answer 1

The easiest way I see to do this is to rotate it first so its axis of symmetry is vertical (or horizontal), do the calculations, and rotate it back.

The transformation for rotation is best described as
x → xcosθ - ysinθ
y → xsinθ + ycosθ where θ is the angle of rotation

So rotate by 45° anticlockwise whence axis becomes vertical:

(10, 0)
x → xcos45° - ysin45° = 10cos45° - 0 = 5√2
y → xsin45° + ycos45° = 10sin45° = 5√2

(-2,0)
x → xcos45° - ysin45° = -2cos45° = -√2
y → xsin45° + ycos45° = -2sin45° = -√2

(12,4)
x → x cos45° - ysin45° = 12cos45° - 4sin45° = 4√2
y → xsin45° + ycos45° = 12sin45° + 4cos45° = 8√2

So now we find the equation of the parabola with a vertical exis through (5√2, 5√2), (-2√2, -2√2) and (4√2, 8√2)

Well the parabola is of the form y = ax² + bx + c

When x = 5√2, y = 5√2
So 50a + 5√2b + c = 5√2 ................. (1)

When x = -√2, y = -√2
So 2a - √2b + c = -√2................. ...... (2)

And when x = 4√2, y = 8√2
So 32a + 4√2b + c = 8√2 ................. (3)

Equation (1) + 5 x equation (2)

60a + 6c = 0
ie 10a + c = 0 .................... ........... (4)

Equation (3) + 4 x equation (2)

40a + 5c = 4√2 ................ ........... (5)

Equation (5) - 5 x equation (4)

-10a = 4√2
a = -0.4√2
Thus c = 4√2
and b = 4.2

So the rotated parabola is y = -0.4√2x² + 4.2x + 4√2 ← ***

Now we need to unrotate this parabola and to do that we need to rotate it BACK by an angle of 45° ie an angle of -45°

Under this rototation

x → xcos(-45°) - ysin(-45°) = x/√2 + y/√2
y → xsin(-45°) + ycos(-45°) = -x/√2 + y/√2

Now y = -0.4√2x² + 4.2x + 4√2
Thus -x/√2 + y/√2 = -0.4√2(x/√2 + y/√2)² + 4.2(x/√2 + y/√2) + 4√2
= -0.4√2(x²/2 + 2xy/2 + y²/2) + 4.2x/√2 + 4.2y/√2 + 4√2
= -0.2√2x² - 0.4√2xy - 0.2√2y² + 4.2x/√2 + 4.2y/√2 + 4√2
Multiply by √2
-x + y = -0.4x² - 0.8xy - 0.4y² + 4.2x + 4.2y + 8

Multiply by 10 and collect terms:

4x² + 8xy + 4y² - 52x - 32y - 80 = 0
ie x² + 2xy + y² - 13x - 8y - 20 = 0

no sorry made another .... whoops!!!
It makes no sense ... it is correct up to line marked ← ***

Answer 2

I believe the equation is x^2 -8x - 20 = 7y. I didn't use the slope information you gave. I just solved it based on the three points. Using the general equation for a parabola, y = Ax^2 + Bx + C, you can just plug in the three pairs of (x,y) values, which gives you three equations and three variables, and then you solve for A, B, and C. This gives you A = 1/7, B = -8/7, and c = -20/7.

Answer 3

take the eq as
x^2+y^2+2gx+2fy+c=0
now put(10,0)
100+20g+c=0
put(-2,0)
4-8g+c=0
solve these for g&c
put third point
144+16+24g+8f+c=0
use the values of g&c to find f
use all these values in
x^2+y^2+2gx+2fy+c=0
to get the eq.