Slope Form a Line

Q. Find the coordinates of the foci, the vertices, the lenghts of major and minor axes and the eccentricity of the ellipse $9x^2+4y^2=36$

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

Q. Let $\alpha ,\beta$ are the angle of inclination of the tangents to the axis of the parabola $y^2=4ax$ drawn from the point $P.$

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}\cot \alpha \cot \beta =k,\text{then locus of}P\text{is}& \text{(P)}& kx=a\\ \text{(B)}& \text{If}\tan \alpha +\tan \beta =k,\text{then locus of}P\text{is}& \text{(Q)}& y=k(x-a)\\ \text{(C)}& \text{If}\tan (\alpha +\beta )=k,\text{then locus of}P\text{is}& \text{(R)}& kx=y\\ \text{(D)}& \text{If}\tan \alpha \tan \beta =k,\text{then locus of}P\text{is}& \text{(S)}& xy=k\\ & & \text{(T)}& x=ka\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(A)}\to \text{(T)},\text{(B)}\to \text{(R)},\text{(C}\to \text{(Q)},\text{(D)}\to \text{(P)}$
2. $\text{(A)}\to \text{(R)},\text{(B)}\to \text{(Q)},\text{(C)}\to \text{(P)},\text{(D)}\to \text{(T)}$
3. $\text{(A)}\to \text{(P)},\text{(B)}\to \text{(Q)},\text{(C)}\to \text{(Q)},\text{(D)}\to \text{(S)}$
4. $\text{(A)}\to \text{(S)},\text{(B)}\to \text{(Q)},\text{(C)}\to \text{(P)},\text{(D)}\to \text{(T)}$

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

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 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.

When $\tan x=1$, what does $x$ equal?

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

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. If a focal chord to $y^2=16x$ is tangent to $(x-6{)}^2+y^2=2$, then the possible value(s) of the slope of this chord is/are

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