Normals are drawn from a point P(h, k) with slopes $m_1,m_2,m_3$ to the parabola $C_1:y^2=4x$

If the curve C is the locus of point P with $m_1{m}_2=2$, then number of common tangents to the curve $C_1$ and curve C is

Q. Normals are drawn from the point $P$ with slopes $m_1,m_2,m_3$ to the parabola $y^2=4x.$ If the locus of $P$ when $m_1,m_2=\alpha$ is part of the parabola itself then the value of $\alpha$ is.

Q.

Normals are drawn from a point P(h, k) with slopes $m_1,m_2,m_3$ to the parabola $C_1:y^2=4x$

${\sum }_{\alpha =0}^{\infty }(m_1{)}^{\alpha }+{\sum }_{\alpha =0}^{\infty }(m_2{)}^{\alpha }+{\sum }_{\alpha =0}^{\infty }(m_3{)}^{\alpha }=\frac{89}{22},where|m_i|<1\forall i=1,2,3then67h-89k=$

Q. Three normals with slopes $m_1,m_2$ and $m_3$ are drawn from a point $P$ not on the axis of the parabola $y^2=4x.$ If $m_1{m}_2=\alpha$, results in the locus of $P$ being a part of the parabola, then the value of $\alpha$ is

Q. Consider two concentric circles $C_1:x^2+y^2-4=0$ and $C_2:x^2+y^2-9=0.$ A parabola is drawn through the points where $C_1$ meets $y-$axis and having an arbitrary tangent of $C_2$ as its directrix. If $C$ is the curve of locus of focus of drawn parabola and $e$ is the eccentricity of the curve $C,$ then the value of $12e$ is

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