Dr. SK Goyal Solutions for Chapter: The Straight Lines, Exercise 7: EXERCISE ON LEVEL-I

Two consecutive sides of a parallelogram are $4x+5y=0$ and $7x+2y=0$. If the equation of one diagonal is $11 x+7y=9$, find the equation of other diagonal.

Equation of the base of an equilateral triangle is $3x+4y=9$ and its vertex is at the point $\left(1,2\right)$. Find the equations of the other sides and the length of each side of the triangle.

Show that the straight line $ax+by+c=0$ divides the join of points $A\left(x_1,y_1\right)$ and $B\left(x_2,y_2\right)$ in the ratio $-\frac{a{x}_1+b{y}_1+c}{a{x}_2+b{y}_2+c}$.

Explain trhe negative sign.

Find equations of the sides of the triangle having $\left(3,-1\right)$ as a vertex, $x-4y+10=0$ and $6 x+10y-59=0$ being the equation of an angle bisector and a median respectively drawn from different vertices.

The base $BC$ of a triangle $ABC$ contains the points $P\left(p_1,q_1\right)$ and $Q\left(p_2,q_2\right)$ and the equation of sides $AB$ and $AC$ are $p_{1}x+q_{1}y=1$ and $q_{2}x+p_{2}y=1$ respectively. Prove that the equations of $\mathrm{AP}$ and $\mathrm{AQ}$ are

$\left(p_1{q}_2+q_1{p}_2-1\right)\left(p_{1}x+q_{1}y-1\right)=\left(p_1^2+q_1^2-1\right)\left(q_{2}x+p_{2}y-1\right)$

and $\left(2p_2{q}_2-1\right)\left(p_{1}x+q_{1}y-1\right)=\left(p_1{p}_2+q_1{q}_2-1\right)\left(q_{2}x+p_{2}y-1\right)$ respectively.

Equations of two straight lines are $x\cos \alpha +y\sin \alpha =p$ and $x\cos \beta +y\sin \beta =p^{\text{'}}$. Show that the area of the quadrilateral formed by the two lines and the perpendiculars drawn from the origin to the lines is $\left|\frac{1}{2\sin \left(\beta -\alpha \right)}\left\{2p{p}^{\text{'}}-\left(p^2+{\left(p^{\text{'}}\right)}^2\right)\cos \left(\alpha -\beta \right)\right\}\right|$.

One diagonal of a square is the portion of the line $7 x+5y=35$ intercepted by the axes. Obtain the extremities of the other diagonal.

If $p,x_1,x_2,\dots ,x_i,\dots$ and $q,y_1,y_2,\dots ,y_i,\dots$ form two infinite arithmetic sequences with common difference $a$ and $b$ respectively, then find the locus of the point $\left(\alpha ,\beta \right)$, where

$\alpha =\frac{1}{n}\sum _{i=1}^n{x}_i$ and $\beta =\frac{1}{n}\sum _{i=1}^n{y}_i.$