General Equation of Parabola

Q. 8.Focus (0-3); directrix y-3

Q. If the parabola $y=(a-b)x^2+(b-c)x+(c-a)$ touches the x - axis then the line ax + by + c = 0

1. always has negative slope
2. always perpendicular to x-axis
3. Always passes through a fixed point
4. represents the family of parallel lines

Q.

The vertices of a triangle are A(10, 4), B(-4, 9), and (-2, -1). Find the equation of its altitude which passes through (10, 4)

1. x - 5y + 10 = 0

2. 5x - y + 20 = 0

3. 5x - y + 10 = 0

4. x - 5y + 20 = 0

Q. The point on the parabola $y^2=18x$, for which the ordinate is three times the abscissa, is

1. (6, 2)
2. (-2, -6)
3. (3, 18)
4. (2, 6)

Q. If the line $y=mx+a$ meets the parabola $y^2=4ax$ at two points whose abscissas are $x_1$ and $x_2,$ then $x_1+x_2=0$ if

1. $m=-1$
2. $m=1$
3. $m=2$
4. $m=-\frac{1}{2}$

Q.

Equation of the parabola whose axis is $y=x$, distance from origin to vertex is $\sqrt{2}$ and distance from origin to focus is $2\sqrt{2},$ is (Focus and vertex lie in 1st quadrant) :

1. $(x+y{)}^2=2(x+y-2)$

2. $(x-y{)}^2=8(x+y-2)$

3. $(x-y{)}^2=4(x+y-2)$

4. $(x+y{)}^2=4(x+y-2)$

Q.

How many of the following statements are correct ___

1. A conic with eccentricity equal to one is called a parabola

2. If $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents a parabola, then

$abc+2fgh-a{f}^2-b{g}^2-c{h}^2\ne 0and{h}^2=ab$

Q. If equation of the parabola is $(6x+8y-5{)}^2=60(8x-6y+9),$ then

1. Equation of axis of the parabola is $6x+8y-5=0$
2. Equation of tangent at vertex of the parabola is $6x+8y-5=0$
3. Length of latus rectum is $\frac{3}{2}$ units
4. Length of latus rectum is $6$ units

Q. The equation of the parabola whose axis is vertical and passes through the points (0, 0), (3, 0) and (-1, 4) is

1. $x^2-3x-y=0$
2. $x^2+3x+y=0$
   
3. $x^2-4x-2y=0$
   
4. $x^2-4x-2y=0$

Q. The equation of the parabola whose vertex is $(-3,0)$ and directrix is $x+5=0,$ is

1. $x^2=8(y+3)$
2. $y^2=8(x+3)$
3. $x^2=8(y-3)$
4. $y^2=8(x-3)$

Q. $y=f\left(x\right)$ is the parabola of the form $y=x^2+ax+1,$ its tangent at the point of intersection of $y-$axis and parabola also touches the circle $x^2+y^2=r^2.$ It is known that no point of the parabola is below $x-$axis. The radius of the circle (in units) when $a$ attains its maximum value is

1. $\frac{1}{\sqrt{10}}$
2. $\frac{1}{\sqrt{5}}$
3. $1$
4. $\sqrt{5}$