Straight Lines & Pair of Straight Lines - Math PYQ

If the orthocenter of the triangle formed by the lines \(2 x+3 y-1=0, x+2 y+1=0\) and \(a x+b y-1=0\) lies at origin, then \(\frac{1}{a}+\frac{1}{b}\) is equal to

The equation \(8 x^2-24 x y+18 y^2-6 x+9 y-5=0\) represents a

Find the equation of a line passing through the point \((4,3)\), which cuts a triangle of minimum area from the first quadrant.

If \(A(4,7), B(-7,8)\) and \(C(1,2)\) are the vertices of \(\triangle A B C\), then the equation of perpendicular bisector of the side \(A B\) is

The ratio in which the straight line \(3 x+4 y=6\) divides the join of the points \((2,-1)\) and \((1,1)\) is

The angle between the lines represented by \(\cos \theta(\cos \theta+1) x^2-\left(2 \cos \theta+\sin ^2 \theta\right) x y+(1-\cos \theta) y^2=0\) is

Find the angle between the pair of lines represented by the equation \(x^2+4 x y+y^2=0\).

If the acute angle between lines \(a x^2+2 h x y+b y^2=0\) is \(\frac{\pi}{4}\), then \(4 h^2\) is equal to

If the axes are rotated through an angle \(45 \Upsilon\), the coordinates of the point \((2 \sqrt{2},-3 \sqrt{2})\) in the new system are

the sum of the squares of the intercepts made the line \(5x-2y=10\) on the coordinate axes equals

If \(m_1\) and \(m_2,\left(m_1>m_2\right)\) are the slopes of the lines represented by \(5 x^2-8 x y+3 y^2=0\), then \(m_1: m_2\) equals

The equation of the pair of straight lines perpendicular to the pair \(2 x^2+3 x y+2 y^2+10 x+5 y=0\) and passing though the origin is

If the centroid of the triangle formed by the lines \(2 y^2+5 x y-3 x^2=0\) and \(x+y=k\) is \(\left(\frac{1}{18}, \frac{11}{18}\right)\), then the value of \(k\) equals

If \(2x+3y+4=0\) is the perpendicular bisector of the line segment joining the points A(1, 2) and B(\(\alpha,\beta\)), then the value of \(13\alpha+13\beta\) equals

For three consecutive odd integers \(a \cdot b\) and \(c\), if the variable line \(a x+b y+c=0\) always passes through the point \((\alpha, \beta)\), the value of \(\alpha^2+\beta^2\) equals

If the slope of one of the lines represented by \(a x^2+2 h x y+b y^2=0\) is the square of the other then, \(\left|\frac{a+b}{h}+\frac{8 h^2}{a b}\right|\) is equal to

The value of \(p\) for which the equation \(x^2+p x y+y^2-5 x-7 y+6=0\) represents a pair of straight lines is

If \(A(1,3)\) and \(C(7,5)\) are two opposite vertices of a square, then find the equation of a side passing through \(A\).

The distance of the point \((1,2)\) from the line \(x+y+5=0\) measured along the line parallel to \(3 x-y=7\) is equal to

Find the equation of a line which passes through \(\left(2 \cos ^3(\theta), 2 \sin ^3(\theta)\right)\) and is perpendicular to the line \(x \cos (\theta)-y \sin (\theta)=2 \cos (2 \theta)\).