Straight Lines

Let ABC be a triangle with A (-3, 1) and $\angle ACB=\theta ,$ $0<\theta <\frac{\pi }{2}.$ <!-- [if gte mso 9]><xml> </xml><![endif]-->  <!-- [if gte mso 9]><xml> </xml><![endif]--> If the equation of the median through B is 2x + y – 3 = 0 and the equation of angle bisector of C is 7x – 4y – 1 = 0, then tanq is equal to:

A 10 inches long pencil AB with mid point C and a small eraser P are placed on the horizontal top of a table such that PC = $\sqrt[]{5}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> inches and <!-- [if gte mso 9]><xml> </xml><![endif]-->  $\angle PCB=ta{n}^{-1}\left(2\right).$

The acute angle through which the pencil must be rotated about C so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is:

Let A(, 2), B(, 6) and $C\left(\frac{\alpha }{4},-2\right)$ be vertices of a $\Delta ABC.$ If $\left(5,\frac{\alpha }{4}\right)$ is the circumcentre of $\Delta ABC,$ then which of the following is NOT correct about $\Delta ABC?$

Let m₁, m₂ be the slopes of two adjacent sides of a square of side a such that a² + 11a + $3\left(m_1^2+m_2^2\right)=220.$ If one vertex of the square is $\left(10\left(cos\alpha -sin\alpha \right),10\left(sin\alpha +cos\alpha \right)\right),$ where $\alpha \in \left(0,\frac{\pi }{2}\right)$ and the equation of one diagonal is $\left(cos\alpha -sin\alpha \right)x+\left(sin\alpha +cos\alpha \right)y=10,$ then $72\left(si{n}^4\alpha +co{s}^4\alpha \right)+a^2-3a+13$ is equal to

A circle touches both the y-axis and the line x + y = 0. Then the locus of its center is:

Let A(1, 1) B(-4, 3), C(-2, -5) be vertices of a triangle ABC, P be a point on side BC, and ${\Delta }_{1}and{\Delta }_2$  be the area of triangles APB and ABC, respectively. If ${\Delta }_1:{\Delta }_2=4:7,$ <!-- [if gte mso 9]><xml> </xml><![endif]--> then the area enclosed by the lines AP, AC and the x-axis is

Let

A₁ = $\left\{\left(x,y\right):\left|x\right|\le y^2,\left|x\right|+2y\le 8\right\}and$ <!-- [if gte mso 9]><xml> </xml><![endif]--> 

A₂ = $\left\{\left(x,y\right):\left|x\right|+\left|y\right|\le k\right\}:$ <!-- [if gte mso 9]><xml> </xml><![endif]--> If 27 (Area A₁) = 5 (Area A₂), then k is equal to:

Let $\frac{dy}{dx}=\frac{ax-by+a}{bx+cy+a},$ <!-- [if gte mso 9]><xml> </xml><![endif]--> where a, b, c, are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the point (11, 6) from this circle is:      

Let A(a, -2), B(a, 6) and $C\left(\frac{\alpha }{4},-2\right)$ be vertices of a $\Delta ABC.$ <!-- [if gte mso 9]><xml> </xml><![endif]--> If $\left(5,\frac{\alpha }{4}\right)$ <!-- [if gte mso 9]><xml> </xml><![endif]--> is the circumcentre of $\Delta ABC,$ <!-- [if gte mso 9]><xml> </xml><![endif]--> then which of the following is NOT correct about $\Delta ABC?$ <!-- [if gte mso 9]><xml> </xml><![endif]--> 

Let m₁, m₂ be the slopes of two adjacent sides of a square of side a such that a² + 11a + $3\left(m_1^2+m_2^2\right)=220.$ <!-- [if gte mso 9]><xml> </xml><![endif]--> If one vertex of the square is $\left(10\left(cos\alpha -sin\alpha \right),10\left(sin\alpha +cos\alpha \right)\right),$ <!-- [if gte mso 9]><xml> </xml><![endif]--> where $\alpha \in \left(0,\frac{\pi }{2}\right)$ <!-- [if gte mso 9]><xml> </xml><![endif]--> and the equation of one diagonal is $\left(cos\alpha -sin\alpha \right)x+\left(sin\alpha +cos\alpha \right)y=10,$ then $72\left(si{n}^4\alpha +co{s}^4\alpha \right)+a^2-3a+13$ <!-- [if gte mso 9]><xml> </xml><![endif]--> is equal to: