Visualizing Conics from Right Circular Cone

Q.

If the locus of the mid-point of the line segment from the point $(3,2)$ to a point on the circle, $x^2+y^2=1$ is a circle of the radius $r$, then $r$ is equal to

1. $\frac{1}{4}$

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

3. $1$

4. $\frac{1}{3}$

Q. The point of intersection of the straight lines represented by $6x^2+xy-40y^2-35x-83y+11=0$ is :

1. $(0,\frac{-11}{5})$
2. $(3,-1)$
3. $(\frac{11}{2},0)$
4. None of the above

Q. The equation
$25[x^2+y^2-2x+1]=(4x-3y+1{)}^2$ represents

1. Parabola
2. Circle
3. Pair of straight line
4. None of these

Q. The equation $\lambda x^2+4xy+y^2+\lambda x+3y+2=0$ represents

1. a parabola if $\lambda =4$
2. an ellipse if $\lambda >4$
3. a hyperbola if $\lambda <4$
4. a pair of straight lines if $\lambda =2$

Q. A point moves so that the sum of the squares of its distances from the six faces of a cube given by $x=\pm 1,y\pm 1,z=\pm 1$ is 10 sq. units. Locus of the point is

1. $x^2+y^2+z^2=2$
   
2. $x^2+y^2+z^2=1$
   
3. x + y + z = 1
   
4. x + y + z = 2

Q. The focii of the parabolas $y^2=4x,x^2=4y,y^2+4x=0,x^2+4y=0$ taken in order will form the vertices of a

1. rectangle
2. parallelogram
3. rhombus
4. square

Q. Consider a parabola $S:y^2=4x$. Let points A(–1, 0) and B(0, 1) and F be the focus of parabola S.
Let Q(13, 9) be a given fixed point and $P(\alpha ,\beta )$ be a point on the parabola 'S' such that PQ + PF is least, then

1. $\alpha =3$
2. $\alpha +\beta =15$
3. $\alpha -\beta =3$
4. $\beta =6$

Q. The perpendicular distance (in units) of the plane $x-2y-3z-3\sqrt{14}=0$ from origin is equal to

Q. The locus of the point $x=t+\frac{1}{t}$ and $y=t-\frac{1}{t}$ is

1. a straight line
2. a circle
3. a hyperbola
4. a parabola

Q. The radius of the cirele in which the sphere \vert r\vert=5 is cut by the plane r. (i+j+k)=3\surd3 is=

Q. Find the equation of hyperbola whose focus is $(a,0)$, directrix is $2x-y+a=0$ and eccentricity $=\frac{4}{3}$

Q. The equation of plane whose perpendicular distance from origin is $\sqrt{17}\text{units}$ and normal vector to plane is $2\hat{i}-3\hat{j}-2\hat{k},$ is

1. $2x-3y-2z=\sqrt{17}$
2. $2x-3y-2z=17$
3. $-2x+3y-2z=\sqrt{17}$
4. $2x-3y+2z=17$

Q. The equation of plane whose perpendicular distance from origin is $2\text{units}$ and vector $\hat{i}-2\hat{j}-2\hat{k}$ is normal to the plane, is

1. $x-2y-2z=2$
2. $x-2y-2z=3$
3. $x-2y-2z=6$
4. $x+2y+2z=6$

Q.

Who named the different conic?

Q. A circle and a parabola intersect at four points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ and $(x_4,y_4)$. Then $y_1+y_2+y_3+y_4$ is equal to

1. $4$
2. $3/2$
3. $2$
4. $0$

Q.

Match the following.

In column 1, we are given a two dimensional projection of a right circular cone and a plane with dotted line. Match them with resulting conic given in column 2.

1. P - 2, Q - 1, S - 4, R - 3

2. P - 1, Q - 2, S - 3, R - 4

3. P - 2, Q - 1, R - 3, S - 4

4. P - 1, Q - 2, S - 3, R - 4

Q. Find the co-ordinates of the point from which tangents drawn to the circle $x^2+y^2–6x–8y+3=0$ such that the mid point of its chord of contact is $(1,1)$.

Q.

Match the following.

In column 1, we are given a two dimensional projection of a right circular cone and a plane with dotted line. Match them with resulting conic given in column 2.

1. P - 2, Q - 1, S - 4, R - 3

2. P - 1, Q - 2, S - 3, R - 4

3. P - 1, Q - 2, S - 3, R - 4

4. P - 2, Q - 1, R - 3, S - 4

Q. Number of intersecting points of the conic $4x^2+9y^2=1$ and $4x^2+y^2=4$ is

1. $1$
2. $2$
3. $3$
4. $0$ (zero)

Q. The equation $\lambda x^2+4xy+y^2+\lambda x+3y+2=0$ represents

1. a parabola if $\lambda =4$
2. an ellipse if $\lambda >4$
3. a hyperbola if $\lambda <4$
4. a pair of straight lines if $\lambda =2$

Q. Statement$-1$ : If side $BC$ and the ratio $\frac{r_2}{r_3}$ of an acute angled triangle is given, then the locus of point $A$ is a hyperbola.

Statement$-2$: If the base of a triangle is given and difference of two variable sides is constant (less than the base), then locus of variable vertex is a hyperbola .

1. Both statements are true and statement $-2$ is the correct explanation of statement $-1$.
2. Both statements are true but statement $-2$ is NOT the correct explanation of statement $-1$.
3. Statement$-1$ is true and statement $-2$ is false
4. Statement$-1$ is false and statement $-2$ is true.

Q. The diagram is drawn on a Cartisian plane.
PQRS is a square. The coordinates of $S$ are:

1. $(-3,3)$
2. $(3,-3)$
3. $(-3,-3)$
4. $(-3,2)$

Q. The eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ which

​​​​​​​passes through the points $(3,0)$ and $(3\sqrt{2},2)$ is $\underset{–}{}$

Q. The locus of the point $x=t+\frac{1}{t}$ and $y=t-\frac{1}{t}$ is

1. a straight line
2. a circle
3. a parabola
4. a hyperbola

Q. Find the direction cosines of the unit vector perpendicular to the plane $\overrightarrow{r}.(6\hat{i}-3\hat{j}-2k)+1=0$ passing through the origin.