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Straight Lines and Pair of Straight LinesCircleParabolaEllipseHyperbola
Straight Lines and Pair of Straight Lines
Practice Questions
MCQ (Single Correct Answer)
1
The locus of the mid-point of the portion of the line $x \cos \alpha+y \sin \alpha=p$ intercepted by the coordinate axes, where $p$ is a constant, is
AP EAPCET 2024 - 23th May Morning Shift
2
The origin is shifted to the point $(2,3)$ by translation of axes and then the coordinate axes are rotated about the origin through an angle $\theta$ in the counter - clockwise sense. Due to this if the equation $3 x^2+2 x y+3 y^2-18 x-22 y+50=0$ is transformed to $4 x^2+2 y^2-1=0$, then the angle $\theta$ is euqal to
AP EAPCET 2024 - 23th May Morning Shift
3
If the straight line passing through $P(3,4)$ makes an angle $\frac{\pi}{6}$ with the positive $X$-axis in anti-clockwise direction and meets the line $12 x+5 y+10=0$ at $Q$, then the length of the segment $P Q$ is
AP EAPCET 2024 - 23th May Morning Shift
4
The equation of the perpendicular bisectors of the sides $A B$ and $A C$ of $\triangle A B C$ are $x-y+5=0$ and $x+2 y=0$ respectively, If the coordinates of $A$ are $(1,-2)$, then the equal of the line $B C$ is
AP EAPCET 2024 - 23th May Morning Shift
5
A pair of lines drawn through the origin forms a right angled isosceles triangle with right angle at the origin with the line $2 x+3 y=6$. The area (in sq units) of the triangle thus formed is
AP EAPCET 2024 - 23th May Morning Shift
6
The combined equation of the bisectors of the angles between the lines joining the origin to the points of intersection of the curve $x^2+y^2+x y+x+3 y+1=0$ and the line $x+y+2=0$ is
AP EAPCET 2024 - 23th May Morning Shift
7
The locus of a variable point which forms a triangle of fixed area with two fixed points is
AP EAPCET 2024 - 22th May Evening Shift
8
$A$ line $L$ passing through the point $P(-5,-4)$ cuts the lines $x-y-5=0$ and $x+3 y+2=0$ respectively at $Q$ and $R$ such that $\frac{18}{P Q}+\frac{15}{P R}=2$, then slope of line $L$ is
AP EAPCET 2024 - 22th May Evening Shift
9
If the reflection of a point $A(2,3)$ in $X$-axis is $B$, reflection of $B$ in the line $x+y=0$ is $C$ and the reflection of $C$ in $x-y=0$ is $D$, then the point of intersection of the lines $C D, A B$ is
AP EAPCET 2024 - 22th May Evening Shift
10
The equation of a line which makes an angle of $45^{\circ}$ with each of the pair of lines $x y-x-y+1=0$ is
AP EAPCET 2024 - 22th May Evening Shift
11
If the slope of one of the lines in the pair of lines $8 x^2+a x y+y^2=0$ is thrice the slope of the second line, then $a$ is equal to
AP EAPCET 2024 - 22th May Evening Shift
12
The equation of the locus of points which are equidistant from the point $(2,3)$ and $(4,5)$ is
AP EAPCET 2024 - 22th May Morning Shift
13
The equation of the side of an equilateral triangle is $x+y=2$ and one vertex is $(2,-1)$. The length of the side is
AP EAPCET 2024 - 22th May Morning Shift
14
The orthocentre of the triangle formed by lines $x+y+1=0, x-y-1=0$ and $3 x+4 y+5=0$ is
AP EAPCET 2024 - 22th May Morning Shift
15

If the slope of one of the pair of lines represented by $2 x^2+3 x y+K y^2=0$ is 2 , then the angle between the pair of lines is

AP EAPCET 2024 - 22th May Morning Shift
16
The length of $x$-intercept made by pair of lines $2 x^2+x y-6 y^2-2 x+17 y-12=0$ is
AP EAPCET 2024 - 22th May Morning Shift
17
Suppose the axes are to be rotated through an angle $\theta$ so as to remove the $x y$ form from the equation $3 x^2+2 \sqrt{3} x y+y^2=0$. Then, in the new coordinate system the equation $x^2+y^2+2 x y=2$ is transformed to
AP EAPCET 2024 - 21th May Evening Shift
18
$P$ is a point on $x+y+5=0$, whose perpendicular distance from $2 x+3 y+3=0$ is $\sqrt{13}$, then the coordinates of $P$ are
AP EAPCET 2024 - 21th May Evening Shift
19
For $\lambda, \mu \in R,(x-2 y-1)+\lambda(3 x+2 y-11)=0$ and $(3 x+4 y-11)+\mu(-x+2 y-3)=0$ represent two families of lines. If the equation of the line common to both the families is $a x+b y-5=0$. Then, $2 a+b=$
AP EAPCET 2024 - 21th May Evening Shift
20
If the pair of lines represented by $3 x^2-5 x y+P y^2=0$ and $6 x^2-x y-5 y^2=0$ have one line in common, then the sum of all possible value of $P$ is
AP EAPCET 2024 - 21th May Evening Shift
21
$P$ is a variable point such that the distance of $P$ from $A$ $(4,0)$ is twice the distance of $P$ from $B(-4,0)$. If the line $3 y-3 x-20=0$ intersects the locus of $P$ at the points $C$ and $D$, then the distance between $C$ and $D$ is
AP EAPCET 2024 - 21th May Morning Shift
22
When the origin is shifted to $(h, k)$ by translation of axes, the transformed equation of $x^2+2 x+2 y-7=0$ does not contain $x$ term and constant term. Then, $(2 h+k)=$
AP EAPCET 2024 - 21th May Morning Shift
23
Let $\alpha \in R$. If the line $(\alpha+1) x+\alpha y+\alpha=1$ passes through a fixed point $(h, k)$ for all $\alpha$, then $h^2+k^2=$
AP EAPCET 2024 - 21th May Morning Shift
24
The area of the triangle formed by the lines represented by $3 x+y+15=0$ and $3 x^2+12 x y-13 y^2=0$ is
AP EAPCET 2024 - 21th May Morning Shift
25
If all chords of the curve $2 x^2-y^2+3 x+2 y=0$, which subtend a right angle at the origin always passing through the point $(\alpha, \beta)$, then $(\alpha, \beta)=$
AP EAPCET 2024 - 21th May Morning Shift
26
If the origin is shifted to remove the first degree terms from the equation $2 x^2-3 y^2+4 x y+4 x+4 y-14=0$, then with respect to this new coordinate system the transformed equation of $x^2+y^2-3 x y+4 y+3=0$ is
AP EAPCET 2024 - 20th May Evening Shift
27
The circumcentre of the triangle formed by the lines $x+y+2=0,2 x+y+8=0$ and $x-y-2=0$ is
AP EAPCET 2024 - 20th May Evening Shift
28
If the line $2 x-3 y+5=0$ is the perpendicular bisector of the line segment joining $(1,-2)$ and $(\alpha, \beta)$, then $\alpha+\beta=$
AP EAPCET 2024 - 20th May Evening Shift
29
If the area of the triangle formed by the straight lines $-15 x^2+4 x y+4 y^2=0$ and $x=\alpha$ is 200 sq unit, then $|\alpha|=$
AP EAPCET 2024 - 20th May Evening Shift
30
The equation for straight line passing through the point of intersection of the lines represented by $x^2+4 x y+3 y^2-4 x-10 y+3=0$ and the point $(2,2)$ is
AP EAPCET 2024 - 20th May Evening Shift
31
If the origin is shifted to a point $P$ by the translationd axes to remove the $y$-term from the equation $x^2-y^2+2 y-1=0$, then the transformed equation of it is
AP EAPCET 2024 - 20th May Morning Shift
32
A line $L$ intersects the lines $3 x-2 y-1=0$ and $x+2 y+1=0$ at the points $A$ and $B$. If the point $(1,2)$ bisects the line segment $A B$ and $\frac{x}{a}+\frac{y}{b}=1$ is the equation of the line $L$, then $a+2 b+1=$
AP EAPCET 2024 - 20th May Morning Shift
33
A line $L$ passing through the point $(2,0)$ makes an angle $60^{\circ}$ with the line $2 x-y+3=0$. If $L$ makes an acute angle with the positive X-axis in the anti-clockwise direction, then the $Y$-intercept of the line $L$ is
AP EAPCET 2024 - 20th May Morning Shift
34
If the slope of one line of the pair of lines $2 x^2+h x y+6 y^2=0$ is thrice the slope of the other line, then $h=$
AP EAPCET 2024 - 20th May Morning Shift
35

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 $2(a-h+b-g+f-12)=$

AP EAPCET 2024 - 20th May Morning Shift
36
If a variable straight line passing through the point of intersection of the lines $x-2 y+3=0$ and $2 x-y-1=0$ intersects the $X, Y$-axes at $A$ and $B$ respectively, then the equation of the locus of a point which divides the segment $A B$ in the ratio $-2: 3$ is
AP EAPCET 2024 - 19th May Evening Shift
37
Point $(-1,2)$ is changed to $(a, b)$, when the origin is shifted to the point $(2,-1)$ by translation of axes, Point $(a, b)$ is changed to $(c, d)$, when the axes are rotated through an angle of $45^{\circ}$ about the new origin, $(c, d)$ is changed to $(e, f)$, when $(c, d)$ is reflected through $y=x$. Then, $(e, f)=$
AP EAPCET 2024 - 19th May Evening Shift
38
The point $(a, b)$ is the foot of the perpendicular drawn from the point $(3,1)$ to the line $x+3 y+4=0$. If $(p, q)$ is the image of $(a, b)$ with respect to the line $3 x-4 y+11=0$, then $\frac{p}{a}+\frac{q}{b}=$
AP EAPCET 2024 - 19th May Evening Shift
39
A ray of light passing through the point $(2,3)$ reflects on $Y$-axis at a point $P$. If the reflected ray passes through the point $(3,2)$ and $P=(a, b)$, then $5 b=$
AP EAPCET 2024 - 19th May Evening Shift
40
The area (in sq units) of the triangle formed by the lines $6 x^2+13 x y+6 y^2=0$ and $x+2 y+3=0$ is
AP EAPCET 2024 - 19th May Evening Shift
41
If the lines $3 x+y-4=0, x-\alpha y+10=0, \beta x+2 y+4=0$ and $3 x+y+k=0$ represent the sides of a square, then $\alpha \beta(k+4)^2=$
AP EAPCET 2024 - 18th May Morning Shift
42
$A$ is the point of intersection of the lines $3 x+y-4=0$ and $x-y=0$. If a line having negative slope makes an angle of $45^{\circ}$ with the line $x-3 y+5=0$ and passes through $A$, then its equation is
AP EAPCET 2024 - 18th May Morning Shift
43
$2 x^2-3 x y-2 y^2=0$ represents two lines $L_1$ and $L_2$. $2 x^2-3 x y-2 y^2-x+7 y-3=0$ represents another two lines $L_3$ and $L_4$. Let $A$ be the point of intersection of lines $L_1, L_3$ and $B$ be the point of intersection of lines $L_2$ and $L_4$. The area of the triangle formed by lines $A B$. $L_3$ and $L_4$ is
AP EAPCET 2024 - 18th May Morning Shift
44
The area of the triangle formed by the pair of lines $23 x^2-48 x y+3 y^2=0$ with the line $2 x+3 y+5=0$, is
AP EAPCET 2024 - 18th May Morning Shift
45

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

AP EAPCET 2022 - 5th July Morning Shift
46

The length of intercept of $$x+1=0$$ between the lines $$3 x+2 y=5$$ and $$3 x+2 y=3$$ is

AP EAPCET 2022 - 5th July Morning Shift
47

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?

AP EAPCET 2022 - 5th July Morning Shift
48

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

AP EAPCET 2022 - 5th July Morning Shift
49

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

AP EAPCET 2022 - 4th July Evening Shift
50

If the points of intersection of the coordinate axes and $$|x+y|=2$$ form a rhombus, then its area is

AP EAPCET 2022 - 4th July Evening Shift
51

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

AP EAPCET 2022 - 4th July Evening Shift
52

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

AP EAPCET 2022 - 4th July Evening Shift
53

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

AP EAPCET 2022 - 4th July Evening Shift
54

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

AP EAPCET 2022 - 4th July Morning Shift
55

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

AP EAPCET 2022 - 4th July Morning Shift
56

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

AP EAPCET 2022 - 4th July Morning Shift
57

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?

AP EAPCET 2022 - 4th July Morning Shift
58

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

AP EAPCET 2022 - 4th July Morning Shift
59

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

AP EAPCET 2021 - 20th August Morning Shift
60

Find the equation of a straight line passing through $$(-5,6)$$ and cutting off equal intercepts on the coordinate axes.

AP EAPCET 2021 - 20th August Morning Shift
61

Line has slope $$m$$ and $$y$$-intercept 4 . The distance between the origin and the line is equal to

AP EAPCET 2021 - 20th August Morning Shift
62

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

AP EAPCET 2021 - 20th August Morning Shift
63

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

AP EAPCET 2021 - 20th August Morning Shift
64

The acute angle between lines $$6 x^2+11 x y-10 y^2=0$$ is

AP EAPCET 2021 - 20th August Morning Shift
65

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

AP EAPCET 2021 - 20th August Morning Shift
66

The equation of bisector of the angle between the lines represented by $$3 x^2-5 x y+4 y^2=0$$ is

AP EAPCET 2021 - 20th August Morning Shift
67

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

AP EAPCET 2021 - 20th August Morning Shift
68

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

AP EAPCET 2021 - 19th August Evening Shift
69

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

AP EAPCET 2021 - 19th August Evening Shift
70

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

AP EAPCET 2021 - 19th August Evening Shift
71

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

AP EAPCET 2021 - 19th August Evening Shift
72

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

AP EAPCET 2021 - 19th August Evening Shift
73

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

AP EAPCET 2021 - 19th August Evening Shift
74

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

AP EAPCET 2021 - 19th August Evening Shift
75

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

AP EAPCET 2021 - 19th August Evening Shift
76

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

AP EAPCET 2021 - 19th August Morning Shift
77

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

AP EAPCET 2021 - 19th August Morning Shift
78

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

AP EAPCET 2021 - 19th August Morning Shift
79

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

AP EAPCET 2021 - 19th August Morning Shift
80

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

AP EAPCET 2021 - 19th August Morning Shift
81

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

AP EAPCET 2021 - 19th August Morning Shift
82

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

AP EAPCET 2021 - 19th August Morning Shift
83

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

AP EAPCET 2021 - 19th August Morning Shift