Embibe Experts Solutions for Chapter: Parabola, Exercise 4: Exercise-4

Consider the locus of the complex number $z$ in the Argand plane is given by $\mathit{Re}\left(z\right)-2=|z-7+2i|.$ Let $P\left(z_1\right)$ and $Q\left(z_2\right)$ be two complex number satisfying the given locus and also satisfying $\arg \left(\frac{z_1-(2+\alpha i)}{z_2-(2+\alpha i)}\right)=\frac{\pi }{2}, \left(\alpha \in R\right)$ then find the minimum value of $PQ$

Prove that in a parabola the angle $\theta$ that the latus rectum subtends at the vertex of the parabola isindependent of the latus rectum and lies between $\frac{2\pi }{3} \& \frac{3\pi }{4}$.

If $r_1, r_2$ be the length of the perpendicular chords of the parabola $y^2=4ax$ drawn through the vertex, then show that ${\left(r_1{r}_2\right)}^{\frac{4}{3}}=16a^2\left(r_1^{\frac{2}{3}}+r_2^{\frac{2}{3}}\right)$.

Find locus of a point $P$ if the three normals drawn from it to the parabola $y^2=4ax$ are such that two of them make complementary angles with the axis of the parabola.

Prove that the orthocentre of any triangle formed by three tangents to a parabola lies on the directrix.

If tangent drawn at a point $\left(t^2,2t\right)$ on the parabola $y^2=4x$ is same as the normal drawn at a point $\left(\sqrt{5}\cos \varphi ,2\sin \varphi \right)$ on the ellipse $4x^2+5y^2=20$. Find the values of $t \text{\&} \varphi$.

Find the locus of centre of a family of circles passing through the vertex of the parabola $y^2=4ax$, and cutting the parabola orthogonally at the other point of intersection.

Let $A, B, C$ be three points on the parabola $y^2=4ax$. If the orthocentre of the triangle $A B C$ is at the focus then show that the circumcircle of $∆ABC$ touches the $y$-axis.