Embibe Experts Solutions for Chapter: Two Dimensional Analytical Geometry-II, Exercise 1: Exercise

Prove that the line $2x+y=1$ is a tangent to the circle $x^2+y^2+6x-4y+8=0$.

The equation of a circle of radius $6$ units and whose centre lies on the negative direction of $x$-axis at a distance of $4$ units from the origin is 

A focal chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ makes an angle $\theta$ with major axis. Show that its length is $\frac{2a{b}^2}{a^2 {\sin }^2 \theta +b^2 {\cos }^2 \theta }$.

If $PQ$ and $QR$ are two focal chords of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the eccentric angles of $P, Q, R$ are respectively $\alpha , \beta , \gamma$, show that $\tan \frac{\alpha }{2}, \cot \frac{\beta }{2}, \tan \frac{\gamma }{2}$ are in $\mathrm{G}.\mathrm{P}.$.

Show that the tangent to a rectangular hyperbola terminated by its asymptotes is bisected at the point of contact.

A cable of a suspension bridge is in the form of a parabola whose span is $40$ metres. The road way is $5$ metres below the lowest point of the cable. An extra support is provided across the cable $30$ metres above the ground level. Find the length of the support if the height of the pillars are $55$ metres.$$

Find the equation and eccentricity of the ellipse, if the centre of ellipse is same as centre of the hyperbola $4x^2-9y^2+8x-36y-68=0$, length of semi major axis is $3$ units, length of minor axis is $2$ units, the equation of major axis is $x=-1$.

Find the centre, foci and vertices of the hyperbola $16x^2-9y^2-32x-18y+151=0$ and draw the diagram.