Straight Lines and Pair of Straight Lines · Mathematics · AP EAPCET
MCQ (Single Correct Answer)
If the equation of the pair of straight lines passing through the point $(1,1)$ and perpendicular to the pair of lines $3 x^2+11 x y-4 y^2=0$ is $a x^2+2 h x y+b y^2+2 g x+2 f y+12=0$, then $2(a-h+b-g+f-12)=$
Suppose $$P$$ and $$Q$$ lie on $$3 x+4 y-4=0$$ and $$5 x-y-4=0$$ respectively. If the mid-point of $$P Q$$ is $$(1,5)$$, then the slope of the line passing through $$P$$ and $$Q$$ is
The length of intercept of $$x+1=0$$ between the lines $$3 x+2 y=5$$ and $$3 x+2 y=3$$ is
Suppose the slopes $$m_1$$ and $$m_2$$ of the lines represented by $$a x^2+2 h x y+b y^2=0$$ satisfy $$3\left(m_1-m_2\right)-7=0$$ and $$m_1 m_2-2=0$$. Then, which of the following is true?
Suppose that the sides passing through the vertex $$(\alpha, \beta)$$ of a triangle are bisected at right angles by the lines $$y^2-8 x y-9 x^2=0$$. Then, the centroid of the triangle is
Suppose $$P$$ and $$Q$$ are the mid-points of the sides $$A B$$ and $$B C$$ of a triangle where $$A(1,3), B(3,7)$$ and $$C(7,15)$$ are vertices. Then, the locus of $$R$$ satisfying $$A C^2+Q R^2=P R^2$$ is
If the points of intersection of the coordinate axes and $$|x+y|=2$$ form a rhombus, then its area is
Suppose, in $$\triangle A B C, x-y+5=0, x+2 y=0$$ are respectively the equations of the perpendicular bisectors of the sides $$A B$$ and $$A C$$. If $$A$$ is $$(1,-2)$$, the equation of the line joining $$B$$ and $$C$$ is
If the pair of straight lines $$9 x^2+a x y+4 y^2+6 x+b y-3=0$$ represents two parallel lines, then
A line passing through $$P(2,3)$$ and making an angle of $$30^{\circ}$$ with the positive direction of $$X$$-axis meets $$x^2-2 x y-y^2=0$$ at $$A$$ and $$B$$. Then the value of $$P A: P B$$ is
The least distance from origin to a point on the line $$y=x+3$$ which lies at a distance of 2 units from $$(0,3)$$ is
Starting from the point $$A(-3,4)$$, a moving object touches $$2 x+y-7=0$$ at $$B$$ and reaches the point $$C(0,1)$$. If the object travels along the shortest path, the distance between $$A$$ and $$B$$ is
Suppose a triangle is formed by $$x+y=10$$ and the coordinate axes. Then, the number of points $$(x, y)$$ where $$x$$ and $$y$$ are natural numbers, lying inside the triangle is
If the lines represented by $$a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$$ intersect on the $$X$$-axis, which of the following is in general incorrect?
For $$\alpha \in\left[0, \frac{\pi}{2}\right]$$, the angle between the lines represented by $$[x \cos \theta-y] [(\cos \theta+\tan \alpha) x-(1-\cos \theta \tan \alpha) y]=0$$ is
When the axes are rotated through an angle 45$$^\circ$$, the new coordinates of a point P are (1, $$-$$1). The coordinates of P in the original system are
Find the equation of a straight line passing through $$(-5,6)$$ and cutting off equal intercepts on the coordinate axes.
Line has slope $$m$$ and $$y$$-intercept 4 . The distance between the origin and the line is equal to
The equation of the base of an equilateral triangle is $$x+y=2$$ and one vertex is $$(2,-1)$$, then the length of the side of the triangle is
The equation of a straight line which passes through the point $$\left(a \cos ^3 \theta, a \sin ^3 \theta\right)$$ and perpendicular to $$(x \sec \theta+y \operatorname{cosec} \theta)=a$$ is
The acute angle between lines $$6 x^2+11 x y-10 y^2=0$$ is
If the lines, joining the origin to the points of intersection of the curve $$2 x^2-2 x y+3 y^2+2 x-y-1=0$$ and the line $$x+2 y=k$$, are at right angles, then $$k^2$$ equals
The equation of bisector of the angle between the lines represented by $$3 x^2-5 x y+4 y^2=0$$ is
If the bisectors of the pair of lines $$x^2-2 m x y-y^2=0$$ is represented by $$x^2-2 n x y-y^2=0$$, then
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
Find the equation of a line passing through the point $$(4,3)$$, which cuts a triangle of minimum area from the first quadrant.
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 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
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
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
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 $$2x+3y+4=0$$ 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
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 $$m_1$$ 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
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