# Math

posted by .

a parabola whose rootsare at A(-3,0) and B(6,0) has a y-intercept at point C(0,p),p<0, such that angle ABC is a right angle. if the equation of the parabola is y=ax^2+bx=c, compute the value of (abc)^2

• Math -

The roots of the parabola are known (-3 and 6), so the equation of the parabola is in the form:
k(x+3)(x-6)=0 or
k(x²-3x-18)=0 ...(1)
where k is a constant to be determined.

Since the y-intercept is negative, k>0.

If ∠ABC is a right angle, then a circle with AB as diameter intersects the y-axis at intercepts (0,p) and (0,-p) where p<0.

The centre of circle ABC is at ((6-3)/2,0), or (1.5,0), and the radius is (6-(-3))/2 = 4.5.

The equation of circle ABC is therefore:
C1 : (x-1.5)^2+y^2=4.5^2, or
C1: (x-1.5)^2 + y^2 = 20.25

To find intersection with the y-axis, substitute x=0 in C1 and solve for y to get
y=±√18=±3√2

So p=-3√2.

The constant term of equation (1) should equal p (both are y-intercepts), which leads to:
-3√2 = -18k

Solve for k and substitute in (1) to get the equation of the parabola, and hence calculate (abc)^2 as required.

## Respond to this Question

 First Name School Subject Your Answer

## Similar Questions

1. ### math

A parabola has the equation y = 4(x-3)^2-7 Choose 2 true statements: A) The parabola has a minimum value B) The parabola has a maximum value C) The parabola does not cross the y-axis D) The parabola does not cross the x-axis E) The …
2. ### math

A parabola has the equation y = 4(x-3)^2-7 Choose 2 true statements: A) The parabola has a minimum value B) The parabola has a maximum value C) The parabola does not cross the y-axis D) The parabola does not cross the x-axis E) The …
3. ### math

point A(-1,-2) B(7,2) C(k,4) K is a constant. The vertices of ABC. Angle ABC is a right angle. Ive worked out the gradient of AB to be 1/2. How do you calculate the value of k?
4. ### math

point A(-1,-2) B(7,2) C(k,4) K is a constant. The vertices of ABC. Angle ABC is a right angle. Ive worked out the gradient of AB to be 1/2. And the negative reciprocals is 2. How do you calculate the value of k?
5. ### math

point A(-1,-2) B(7,2) C(k,4) K is a constant. The vertices of ABC. Angle ABC is a right angle. Ive worked out the gradient of AB to be 1/2. And the negative reciprocals is -2. How do you calculate the value of k?
6. ### geometry

obtuse triangle ABC with obtuse angle B. Point D on side AC such that angle BDC is also obtuse. Angle DAB is 2/3 of angle ABD. Angle BCD is 1/5 of angle CBD is 4 more than angle BAD. Find the measure of angle ABC.
7. ### Geometry

Given: obtuse triangle ABC with obtuse angle B. Point D on side AC such that angle BDC is also obtuse. Angle DAB is 2/3 of angle ABD. Angle BCD is 1/5 angle BDC and angle CBD is 4 more than angle BAD. What is the measure of angle ABC?
8. ### math

Point A and B of a triangle ABC are at (-1,1) and (3,9), respectively; while point C is on the parabola y=x2. Find coordinate of C so that the area of ABC is minimum, and calculate the largest are of ABC.
9. ### maths

ABC is an acute triangle with \angle BCA = 35 ^\circ. Denote the circumcenter of ABC as O and the orthocenter of ABC as H. If AO=AH, what is the value of \angle ABC (in degrees)?
10. ### Geometry

Please answer at least one . PLEASE 1. Triangle XYZ has a right angle at Y. V,W are on XZ such that XV=XY. WZ=YZ, find ANGLE VYW . 2. STUVWXYZ is a regular octagon. Find SW/ TZ. 3. The interior angles of a convex polygon form an arithmetic …

More Similar Questions