Equation of Parabola When Its Axis Is Parallel to X or Y Axis

Q.

In what ratio the line $y-x+2=0$ divides the line joining the points $(3,-1)$ and $(8,9)$

1. $1:2$

2. $2:1$

3. $2:3$

4. $3:4$

Q.

The straight line $3x+y=9$ divides the line segment joining the points $(1,3)$ and $(2,7)$ in the ratio

1. $3:4$ externally

2. $3:4$ internally

3. $4:5$ internally

4. $5:6$ externally

Q.

If the line $2x+y=k$ passes through the point which divides the segment joining the points $\left(1,1\right)$ and $\left(2,4\right)$ in the ratio $3:2$, then $k$ is equal to

1. $\frac{29}{5}$

2. $5$

3. $6$

4. $\frac{11}{15}$

Q. If tangents are drawn on any focal chord of a parabola, then choose the correct option(s)

1. both the tangents will always be perpendicular to each other
2. extremities of the focal chord and intersection point of the tangents will lie on the circles discribed on the focal chord as diameter
3. both the tangents need not be perpendicular to each other
4. tangents will intersect each other at directrix

Q.

Find the length of the line segment joining the vertex of the parabola $y^2=4ax$ and a point on the parabola where the line-segment makes an angle $\theta$ to the x-axis.

Q. Circles are drawn taking any two focal chords of the parabola $y^2=4ax$ as diameters. If $t_1,t_3$ represent any one end of the different focal chords, then, the equation of the common chord of the circles is

1. $x(t_3+t_1)(t_1^2{t}_3^2+1)=2y{t}_1{t}_3(t_1{t}_3+1)$
2. $x(t_3-t_1)(t_1^2{t}_3^2+1)=2y{t}_1{t}_3(t_1{t}_3+1)$
3. $x(t_3-t_1)(t_1^2{t}_3^2+1)+2y{t}_1{t}_3(t_1{t}_3+1)=0$
4. $x(t_3+t_1)(t_1^2{t}_3^2-1)+2y{t}_1{t}_3(t_1{t}_3+1)=0$

Q. If the $zx-$plane divides the line segment joining $(1,-1,5)$ and $(2,3,4)$ in the ratio $p:1$, then $p+1=$

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

Q. If the locus of the middle points of the normal chords of parabola $y^2=12x$ is given by $6y^2(x–{\lambda }_2)=81{\lambda }_3+y^4$ then

1. ${\lambda }_2=8$
2. ${\lambda }_3=6$
3. ${\lambda }_3=8$
4. ${\lambda }_2=6$

Q. The locus of orthocentre of the triangle formed by three tangents to the parabola $4x^2–8x–3y+10=0$ is

1. $16y=29$
2. $16x=19$
3. $16x=13$
4. $16y=35$

Q.

The point which divides the line segment joining the points $\left(7,-6\right)$ and $\left(3,4\right)$ in the ratio $1:2$ internally lies in the

1. I quadrant

2. II quadrant

3. III quadrant

4. IV quadrant

Q. On a rectangular hyperbola $x^2–y^2=a^2,a>0$, three points $A,B,C$ are taken as follows: $A=(–a,0)$; $B$ and $C$ are placed symmetrically with respect to the x-axis on the branch of the hyperbola not containing $A$. Suppose that the triangle $ABC$ is equilateral. If the side-length of the triangle $ABC$ is $ka$, then $k$ lies in the interval

1. $(0,2]$
2. $(2,4]$
3. $(4,6]$
4. $(6,8]$

Q. A normal chord of the parabola $y^2=4x$ subtending a right angle at the vertex makes an acute angle $\theta$ with the $x-$ axis the angle $\theta$ is equal to

1. ${\cot }^{-1}\sqrt{2}$
2. ${\tan }^{-1}\sqrt{3}$
3. ${\sec }^{-1}\sqrt{3}$
4. ${\tan }^{-1}\sqrt{2}$

Q.

The centroid if the $∆ABC$, where $A=\left(2,3\right),B=\left(8,10\right)$ and $C=\left(5,5\right)$ is

1. $\left(5,6\right)$

2. $\left(6,5\right)$

3. $\left(6,6\right)$

4. $\left(15,18\right)$

Q. If a focal chord of the parabola $y^2=bx$ is $2x-y-8=0$, then the equation of the directrix is

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

Q.

Prove that the line y - x + 2 = 0 divides the join of points (3, -1) and (8, 9) in the ratio 2 : 3.

Q. If the normals drawn at the end points of a variable chord $PQ$ of the parabola $y^2=4ax$ intersect at parabola, then the locus of the point of intersection of the tangent drawn at the points $P$ and $Q$ is

1. $x+a=0$
2. $x-2a=0$
3. $y^2-4x+6=0$
4. $y^2+4x-6=0$

Q. $x-3=t^2,y=4t$ are the parametric equations of the parabola

1. $y^2=16x$
2. $y^2-16x+48=0$
3. $y^2-16x-48=0$
4. $y^2-4x+16=0$

Q. If the $zx-$plane divides the line segment joining $(1,-1,5)$ and $(2,3,4)$ in the ratio $p:1$, then $p+1=$

1. $\frac{1}{2}$
2. $\frac{3}{2}$
3. $\frac{4}{3}$
4. $\frac{1}{3}$

Q. If tangents are drawn on any focal chord of a parabola, then choose the correct option(s)

1. both the tangents will always be perpendicular to each other
2. extremities of the focal chord and intersection point of the tangents will lie on the circles discribed on the focal chord as diameter
3. both the tangents need not be perpendicular to each other
4. tangents will intersect each other at directrix

Q. Circles are drawn taking any two focal chords of the parabola $y^2=4ax$ as diameters. If $t_1,t_3$ represent any one end of the different focal chords, then, the equation of the common chord of the circles is

1. $x(t_3+t_1)(t_1^2{t}_3^2-1)+2y{t}_1{t}_3(t_1{t}_3+1)=0$
2. $x(t_3+t_1)(t_1^2{t}_3^2+1)=2y{t}_1{t}_3(t_1{t}_3+1)$
3. $x(t_3-t_1)(t_1^2{t}_3^2+1)=2y{t}_1{t}_3(t_1{t}_3+1)$
4. $x(t_3-t_1)(t_1^2{t}_3^2+1)+2y{t}_1{t}_3(t_1{t}_3+1)=0$

Q. Prove that the equation to the circle described on the straight line joining the points $(1,{60}^{\circ })$ and $(2,{30}^{\circ })$ as diameter is $r^2-r\left[\cos \right(\theta -{20}^{\circ })+2\cos (\theta -{30}^{\circ }\left)\right]+\sqrt{3}=0$.

Q. If the line segment joining the points $A(a,b)$ and $B(c,d)$ subtends an angle $\theta$ at the origin, then $\cos \theta$ is equal to

1. $\frac{ab+cd}{\sqrt{(a^2+b^2)(c^2+d^2)}}$
2. $\frac{ac+bd}{\sqrt{(a^2+b^2)(c^2+d^2)}}$
3. $\frac{ab+bd}{\sqrt{(a^2+b^2)(c^2+d^2)}}$
4. None of these

Q. If Y-axis is the directrix of the eillipse with eccentricty $e=1/2$ and corresponding focus is at $(3,0)$, then the equation to its auxiliary circle is

1. $x^2+y^2-8x+12=0$
2. $x^2+y^2+8x+12=0$
3. $x^2+y^2-8x+9=0$
4. $x^2+y^2=4$

Q. $x-3=t^2,y=4t$ are the parametric equations of the parabola

1. $y^2=16x$
2. $y^2-16x+48=0$
3. $y^2-16x-48=0$
4. $y^2-4x+16=0$

Q. Any chord $AB$ of $y^2=4ax$ cuts the axis of the parabola at $P$, so that $AP:AB=1:3$ where $A(a{t}_1^2,2a{t}_1),B(a{t}_2^2,2a{t}_2)$, then

1. $t_1{t}_2=-4$
2. $t_1+t_2=0$
3. $2t_1+t_2=0$
4. $t_1+2t_2=0$