Perpendicular lines

[onta...@hotmail.com]

Show that the lines joining the origin to the intersections of the line 4x-y-17 = 0 and the curve 2x^2+2y^2-20x+5y+17 = 0 are perpendicular.

[Avni Pllana]

> Show that the lines joining the origin to the
> intersections of the line 4x-y-17 = 0 and the curve
> 2x^2+2y^2-20x+5y+17 = 0 are perpendicular.

Hi Bill,

The equation of a line through the origin is

y = a*x ,

and together with the other two equations

4x-y-17 = 0 ,

2x^2+2y^2-20x+5y+17 = 0 ,

we obtain the quadratic equation

15*a^2 + 16*a - 15 = 0 .

There are two solutions a1 = 3/5 and a2 = -5/3 , and their product is a1 * a2 = -1 . This means that lines y = a1*x and y = a2*x, are perpendicular.

Best regards,
Avni

[onta...@hotmail.com]

Hi Avni,
I get 15x^2-16xy-15y^2 = 0 and the equations of the lines to be 5x+3y = 0 and 3x-5y = 0.
Because the sum of the coefficients of x^2 and y^2 is zero, the lines are perpendicular.
Regards
Bill