Slope Form a Line

Q. Let a line $y=mx(m>0)$ intersect the parabola $y^2=x$ at a point $P$, other than the origin. The tangent to it at $P$ meet the $x-$ axis at point $Q$. If area $△OPQ=4\text{sq.units}$, then $2m$ is equal to

Q.

The equation of the locus of a point which moves so as to be at equal distances from the point (a, 0) and the $y^{-}$ axis is

1. 

2. 

3. 

4. 

Q. If the line $x+y-1=0$ touches the parabola $y^2=kx,$then the value of $|k|$ is

Q. Two perpendicular chords $AB$ and $AC$ are drawn through the vertex $A$ of the parabola $y^2=4ax$. The locus of circumcentre of the $△ABC$ is $y^2=2ax-\lambda a^2$. Then, the value of $\lambda =$

Q. If $(h,k)$ is a point on the axis of the parabola $2(x-1{)}^2+2(y-1{)}^2=(x+y+2{)}^2$ from where three distinct normals may be drawn, then

1. $h>8$
2. $h>2$
3. $h>4$
4. $h<8$

Q. If two normals to a parabola $y^2=4ax$ intersect at right angles, then the chord joining their feet passes through a fixed point whose coordinates are

1. $(-2a,0)$
2. $(a,0)$
3. $(2a,0)$
4. $(-a,0)$

Q.

Three distinct points $\mathrm{A},\mathrm{B}$ and $\mathrm{C}$ given in the two-dimensional coordinate plane such that the ratio of the distance of any one of them from the point $(1,0)$ to the distance from the point $(-1,0)$ is equal to $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$. Then, the circumcenter of the triangle $\mathrm{ABC}$ is at the point

1. $(\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.,0)$

2. $(\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,0)$

3. $(\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.,0)$

4. $\left(0,0\right)$

Q. The equation of the tangent at the vertex of the parabola $x^2+4x+2y=0$ is

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

Q. If from the vertex of the parabola $y^2=4ax$ pair of chords be drawn perpendicular to each other and with these chords as adjacent sides a rectangle is completed then the locus of the vertex of the farther angle of the rectangle is the parabola

1. $y^2=4a(x-4a)$
2. $y^2=4a(x-6a)$
3. $y^2=4a(x-2a)$
4. $y^2=4a(x-8a)$

Q. If from a point $P$, $3$ normals are drawn, to parabola $y^2=4ax$. Then the locus of $P$ such that three normals intersect the $x-$ axis at points whose distances from vertex are in $A.P.$ is

1. $27a{y}^2=2(x-2a{)}^3$
2. $27a{y}^2=2(x-a{)}^3$
3. $27a{y}^2=2(x+2a{)}^3$
4. $27a{y}^2=2(x+a{)}^3$

Q. The circles described on a focal length(of a parabola) as diameter always touches

1. The tangent at the vertex
2. The axis
3. The directrix
4. Latus rectum

Q. Let $P$ and $Q$ be distinct points on the parabola $y^2=2x$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $△OPQ$ is $3\sqrt{2}\text{sq. units}$, then which of the following is/are the coordinates of $P$?

1. $(4,2\sqrt{2})$
2. $(9,3\sqrt{2})$
3. $(\frac{1}{4},\frac{1}{\sqrt{2}})$
4. $(1,\sqrt{2})$

Q. Normals are drawn from the point $P$ with slopes $m_1\cdot m_2=\alpha$. If $P$ lies on the parabola $y^2=4x$ itself, then $\alpha$ is equal to

Q.

The shortest distance between a circle and an external point is r. The radius of the circle is also 'r'. What is the length of tangent from that point to the circle?

1. r

2. 2r

3. 

4. 

Q. If a tangent to the parabola $y^2=4ax$ makes an angle of $\frac{\pi }{3}$ with the axis of symmetry of the parabola, then point of contact(s) is/are

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

Q. Maximum area of rectangle whose two vertices lies on the $x-$axis & two on the curve $y=3-|x|,\forall |x|<3$

1. $\frac{9}{8}\text{sq. units}$
2. $\frac{9}{4}\text{sq. units}$
3. $3\text{sq. units}$
4. $\frac{9}{2}\text{sq. units}$

Q. If the normals at $A\left(t_1\right)$ and $B\left(t_2\right)$ meet again at $C\left(t_3\right)$ on the parabola $y^2=4ax,$ then the locus of the mid point of $AB$ is

1. $y^2=2ax-a^2$
2. $y^2=2ax+4a^2$
3. $y^2=2ax-4a^2$
4. $y^2=2ax+a^2$

Q. If the ordinates of the points $P$ and $Q$ on the parabola $y^2=12x$ are in the ratio $1:2,$ then the locus of the point of intersection of normals to the parabola at $P$ and $Q$ is

1. $343y^2=-12(x+6{)}^3$
2. $343y^2=12(x-6{)}^3$
3. $343y^2=12(x+6{)}^3$
4. $343y^2=-12(x-6{)}^3$

Q. The normal at the point $(2,3)$ to the circle $x^2+y^2-2x-4y+3=0$ intersects the circle $x^2+y^2=1$ at the points $P$ and $Q$. The area of the circle with $PQ$ as diameter is

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

Q. If the normals at $A\left(t_1\right)$ and $B\left(t_2\right)$ meet again at $C\left(t_3\right)$ on the parabola $y^2=4ax,$ then the locus of the mid point of $AB$ is

1. $y^2=2ax-a^2$
2. $y^2=2ax+4a^2$
3. $y^2=2ax-4a^2$
4. $y^2=2ax+a^2$

Q. Let $x^2=4ky,k>0$, be a parabola with vertex $A$. Let $BC$ be its latus rectum. An ellipse with center on $BC$ touches the parabola at $A$, and cuts $BC$ at points $D$ and $E$ such that $BD=DE=EC$ ($B,D,E,C$ in that order). The eccentricity of the ellipse is

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

Q. Let $S$ be the set of values of paramenter $a$ for which the points of intersection of the parabolas $y^2=3ax$ and $y=\frac{1}{2}(x^2+ax+5)$ are concyclic, then $S$ can be

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

Q. If the line $x+y-1=c$ touches the parabola $x^2+y-x=0,$ then the value of $c$ is

Q. If each pair of equations $a_1{x}^2+b_{1}x+c_1=0,a_2{x}^2+b_{2}x+c_2=0$ and $a_3{x}^2+b_{3}x+c_3=0$ has a common root, then show that
(i) $\frac{c_1{a}_2+c_2{a}_1}{c_1{a}_2-c_2{a}_1}+\frac{a_1{a}_2{b}_3}{a_3(a_1{b}_2-a_2{b}_1)}=0$
(ii)${\left(\frac{a_1{b}_2-a_2{b}_1}{a_1{c}_2-a_2{c}_1}\right)}^2=\frac{a_1{a}_2{c}_3}{c_1{c}_2{a}_3}$

Q. The coordinates of a point on the parabola $y^2=8x$, whose focal distance is $4$ units, is/are

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

Q. If normals are drawn from the point $P$ whose slopes are $m_1$ and $m_2$. If $m_1\cdot m_2=\alpha$ and point $P$ lies on the parabola $y^2=4x$, then the value of $\alpha$ is

Q. If a tangent to a parabola $y^2=4ax$ makes an angle of $\frac{\pi }{3}$ with the axis of the parabola. Then point of contact(s) is/are

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

Q. If the line $x+y-1=0$ touches the parabola $y^2=kx$, then the value of k is

1. $2$
2. $4$
3. $-4$
4. $-2$

Q. The locus of points such that two of the three normals drawn from them to the parabola $y^2=4ax$ coincide is

1. $27a{y}^2=4(x-2a{)}^3$
2. $27ay=4(x-2a{)}^2$
3. $27a{y}^2=4(x-2a)$
4. $27ay=4(x-2a)$

Q. The area of the region bounded by the curves $y=|x-2|,x=1,x=3$ and the x-axis is

1. 1
2. 4
3. 3
4. 2