Vertex

Q.

The graph of a quadratic polynomial is ____?

Q. Three normals are drawn from the point $(7,14)$ to the parabola $x^2-8x-16y=0$. Then the coordinates of the foot of the normal is/are

1. $(0,0)$
2. $(4,-1)$
3. $(-4,3)$
4. $(16,8)$

Q. For the circle $x^2+y^2+6x-8y+9=0,$ which of the following statements is true

1. Circle passes through the point (-3, 4)
2. Circle touches x-axis
3. Circle touches y-axis
4. None of these

Q. The length of the diameter of the circle which touches the $x$ -axis at the point $(1,0)$ and passes through the
point $(2,3)$ is :

1. $\frac{10}{3}$
2. $\frac{6}{5}$
3. $\frac{3}{5}$
4. $\frac{5}{3}$

Q.

If the vertex of parabola y = $x^2$-8x + c lies on x - axis, then the value of c is

1. -16

2. -4

3. 4

4. 16

Q. For the parabola
$x^2+8x+12y+4=0$, which of the following options is/are correct?

1. length of latus rectum is $12$ units
2. coordinates of vertex are $(-4,1)$
3. length of latus rectum is $16$ units
4. coordinates of vertex are $(4,2)$

Q.

The vertex of the parabola $(y-2{)}^2=16(x-1)$ is

1. (1, 2)

2. (-1, 2)

3. (1, -2)

4. (2, 1)

Q. A Circle touches x – axis and cuts off a chord of length 2/from y–axis. The locus of the centre of the circle is

1. A straight line
2. A hyperbola
3. A circle
4. An ellipse

Q.

If the focus of a parabola is (2, 3) and its latus rectum is 8, then the locus of the vertex of the parabola is

1. (0, 3)

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

3. $(y-3{)}^2=8(x-2)$

4. $x-2y+4=0$

Q. Vertex of the parabola $x^2+4x+2y-7=0$ is

1. $(-2,\frac{11}{2})$
2. $(-2,2)$
3. $(-2,11)$
4. $(2,11)$

Q. Graph of $y=e^x\cos x$ is:

1. 
2. 
3. 
4. 

Q. If the area of the triangle whose one vertex is at the vertex of the parabola, $y^2+4(x-a^2)=0$ and the other two vertices are the points of intersection of the parabola and $y$-axis, is $250$ sq. units, then a value of '$a$' is :

1. $5\left(2^{1/3}\right)$
2. $(10{)}^{2/3}$
3. $5$
4. $5\sqrt{5}$

Q. The equation of a circle is $Re\left(z^2\right)+2\left(Im\right(z){)}^2+2Re(z)=0,$ where $z=x+iy$. A line which passes through the centre of the given circle and the vertex of the parabola, $x^2-6x-y+13=0$, has $y-$intercept equal to

Q.

The vertex of the parabola $(y+a{)}^2=8a(x-a)$ is

1. (-a, -a)

2. (a, -a)

3. none of these

4. (-a, a)

Q.

The point on the parabola which is nearest to directrix is

1. End of latusrectum

2. Focus

3. Vertex

4. Centre

Q.

If the focus of a parabola is (-2, 1) and the directrix has the equation x+y=3, then its vertex is

1. (2, -1)

2. $-1,\frac{1}{2}$

3. (0, 3)

4. (-1, 2)

Q.

The vertex of the parabola $x^2+2y=8x-7$is

1. $\left(\frac{9}{2},0\right)$

2. $\left(4,\frac{9}{2}\right)$

3. $\left(2,\frac{9}{2}\right)$

4. $\left(4,\frac{7}{2}\right)$

Q.

If a plane cuts off intercepts OA = a, OB = b, OC = c from the co-ordinate axes, then the area of the triangle ABC=

1. 
   

2. 
   

3. 
   

4. 
   

Q.

The line x-1=0 is the directrix of the parabola $y^2-kx+8=0$. Then one of the values of k is

1. 

2. 

3. 8

4. 4

Q.

Write the coordinates the vertex of the parabola whose focus is at (-2, 1) and directrix is the line x+y-3=0.

Q. The area bounded by the tangent on the curve
$y=4x^2+2x\text{at (0, 0)},y-10=-x\text{and}y=0$ is

1. $\frac{200}{9}\text{sq.units}$
2. $\frac{100}{9}\text{sq.units}$
3. $\frac{100}{3}\text{sq.units}$
4. $20\text{sq.units}$

Q. Find the angle between the prabolas $y^2=4ax$ and $x^2=4by$ at their point of intersection other than origin.

Q. Let two tangents are drawn to the curve $y^2-4(x+y)=3\sin \theta +4\cos \theta -15,x,y,\theta \in R$ from the origin whose slopes are $m_1,m_2$. If the vertex of the curve is at maximum distance from the origin, then the value of $\left|\frac{1}{m_1{m}_2}\right|$ is

Q. Vertex of the parabola formed by taking reflection of $y=4x^2-4x+3$ along the line $y=x$ will be

1. $(\frac{-1}{2},2)$
2. $(\frac{1}{2},-2)$
3. $(\frac{-1}{2},-2)$
4. $(2,\frac{1}{2})$

Q. The vertices of the base of an isosceles triangle lie on the parabola $x^2=4y$ and the base is a part of the line $x=2y-4$. If the third vertex of the triangle lies on y-axis, then

1. one of the vertices of the triangle is $(0,\frac{9}{2})$
2. one of the vertices of the triangle is $(4,4)$
3. Area of the triangle is $15\text{sq. units}$
4. Centroid of the triangle is $(\frac{2}{3},\frac{19}{6})$

Q.

Where is the directrix of a parabola?

Q.

Is the focus always inside the parabola?

Q. The distance from the origin to the normal of the curve $x=2\cos t+2t\sin t,$ $y=2\sin t-2t\cos t$ at $t=\frac{\pi }{4}$ is

Q. Vertex of the parabola $x^2+4x+2y-7=0$ is

1. 
2. 
3. 
4. 

Q. For any quadratic expression $y=a{x}^2+bx+c,a\ne 0$ the $x-$ intercepts will give the roots of $a{x}^2+bx+c=0$.

1. False
2. True