Through the point $ (13,31) $ a straight line is drawn to meet the axes of $ X $ and $ Y $ at $ Q $ and $ S $ respectively. If the rectangle $OQRS$ is completed, the co-ordinates of $ {R} $ satisfy the equation.

$ \frac{31}{x}+\frac{13}{y}=1 $

$ \frac{31}{x}-\frac{13}{y}=1 $

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JEE • MATHEMATICS • STRAIGHT LINES

Straight lines practice · Mixed difficulty

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 421-450 of 555 questions

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421 Easy

421 of 555

Application +4 | -1

The equation of a straight line cutting off a intercept -1 from

y

- axis and being equally inclined to the axes, is -

View answer and explanation

Correct option: A

y-x-1

Option A (

y-x-1

) is correct because a line equally inclined to the axes has a slope of 1 or -1, and with a y-intercept of -1, the equation is y = x - 1, which rearranges to y - x - 1 = 0.

422 Easy

422 of 555

Analyzing +4 | -1

Let

a, b, c

and

d

be non-zero numbers. If the point of intersection of the lines

4 a x+2 a y+c=0

and

5 b x+2 b y+d=0

lies in the fourth quadrant and is equidistant from the two axes, then

View answer and explanation

Correct option: A

3 b c-2 a d=0

Option A (

3bc - 2ad = 0

) is correct because the condition that the intersection point lies in the fourth quadrant and is equidistant from the axes leads to this relationship.

423 Easy

423 of 555

Application +4 | -1

The locus of a point which moves so that the difference of its distances from 2 fixed straight lines which are at right angles is equal to the distance from another fixed straight line is

View answer and explanation

Correct option: A

a straight line

Option A (a straight line) is correct because the locus derived from the given condition results in a linear equation.

424 Easy

424 of 555

Analyzing +4 | -1

Suppose that the points (h, k), (1,2) and (-3, 4) lie on the line

{L_\mathrm{1}}.

If a line

{L_2}

passing through the points (h, k) and (4, 3) is perpendicular to

{L_\mathrm{1}},

then k/ h equals

View answer and explanation

Correct option: C

Option C (3) is correct because by finding the slopes of the lines and using the perpendicularity condition, we can solve for h and k, and then find k/h.

425 Easy

425 of 555

Application +4 | -1

The area of the triangle formed by y-axis, the straight line L passing through (1,1) and (2,0) and the straight line perpendicular to the line L and passing through (1/2,0)

View answer and explanation

Correct option: C

\frac{{25}}{{16}}

sq. units

Option C (

\frac{{25}}{{16}}

sq. units) is correct because it is the area of the triangle calculated using the determinant method with the coordinates of the vertices.

426 Easy

426 of 555

Application +4 | -1

The equation of the base of an equilateral triangle is

x + y = 2

and the vertex is (2,-1) , then the length of the side of the triangle is

View answer and explanation

Correct option: C

\sqrt {2/3}

The correct answer is C (

\sqrt {2/3}

) because it is derived by finding the perpendicular distance from the vertex to the base and using the properties of an equilateral triangle.

427 Easy

427 of 555

Application +4 | -1

Equation of straight line cutting off an intercept 2 from the negative direction of the axes of y and inclined at

{30^ \circ }

to the positive direction of axis of x, is

View answer and explanation

Correct option: D

\sqrt 3 y - x + 2\sqrt 3 = 0

Option D (

\sqrt 3 y - x + 2\sqrt 3 = 0

) is correct because it has a slope of 1/sqrt(3) and a y-intercept of -2.

428 Easy

428 of 555

Application +4 | -1

The polar equation

\cos \mathrm{\theta } + 7\sin \mathrm{\theta } = \frac{1}{\mathrm{r}}

represents a

View answer and explanation

Correct option: C

Straight line

Option C (Straight line) is correct because converting the polar equation to Cartesian coordinates results in a linear equation.

429 Easy

429 of 555

Application +4 | -1

The angle between the lines represented by the equation

\lambda x^2+(1-\lambda)^2 x y-\lambda y^2=0

, is

View answer and explanation

Correct option: D

90^{\circ}

Option D (

90^{\circ}

) is correct because the equation represents perpendicular lines, which is determined by the coefficients of x² and y².

430 Medium

430 of 555

Application +4 | -1

If the lines represented by the equation

2 x^2-3 x y+y^2=0

make angles

\alpha

and

\beta

with

x

-axis, then

\cot ^2 \alpha+\cot ^2 \beta=

View answer and explanation

Correct option: D

5/4

Option D (

5/4

) is correct because it is calculated by finding the sum of squares of cotangents of the angles using the relationship between slopes and coefficients of the given equation.

431 Easy

431 of 555

Application +4 | -1

The angle which the normal to the line

x-\sqrt{3} y+8

=0

passing through the origin, makes with the positive

x

-axis is

View answer and explanation

Correct option: C

120^{\circ}

Option C (

120^{\circ}

) is correct because it is the angle the normal to the line makes with the positive x-axis, derived from the normal form of the line equation.

432 Easy

432 of 555

Application +4 | -1

Find the equation of line, which is parallel to

Y

-axis and at a distance 3 units from left of the origin.

View answer and explanation

Correct option: C

x=-3

Option C (

x=-3

) is correct because a line parallel to the Y-axis has the equation

x = k

, and since it's 3 units to the left of the origin,

k = -3

433 Easy

433 of 555

Application +4 | -1

The equation of the line bisecting perpendicularly the segment joining the points (- 4,6) and (8,8) is

View answer and explanation

Correct option: B

6x + y - 19 = 0

Option B (

6x + y - 19 = 0

) is correct because it represents the equation of the line that is perpendicular bisector of the segment joining the given points.

434 Easy

434 of 555

Application +4 | -1

The equations of two sides of a square whose area is 25 square units are

3x - 4y = 0

and

4x + 3y = 0.

The equations of the other two sides of the square are

View answer and explanation

Correct option: A

3\;x - 4\;y \pm 25 = 0,4\;x + 3\;y \pm 25 = 0

Option A (

3\;x - 4\;y \pm 25 = 0,4\;x + 3\;y \pm 25 = 0

) is correct because it represents the equations of two lines parallel to the given sides of the square and at a distance equal to the side length.

435 Easy

435 of 555

Application +4 | -1

A tangent is drawn to the curve

y=8 / x^2

x y

-plane at the point

A\left(x_0, y_0\right)

, where

x_0=2

, and the tangent cuts the

x

-axis at a point

B

. Then

{A B} \cdot {O B}

is equal to

View answer and explanation

Correct option: D

Option D (3) is correct because the dot product of vectors AB and OB, where B is the x-intercept of the tangent to the curve at point A, equals 3.

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436 Hard

436 of 555

Application +4 | -1

The area of a pentagon whose vertices are

\left( {4,1} \right),\left( {3,6} \right),\left( { - 5,1} \right),\left( { - 3, - 3} \right)

and

\left( { - 3,0} \right)

View answer and explanation

Correct option: A

30 sq. units

The correct answer is A (30 sq. units) because it is the absolute value of the sum of the areas of the triangles formed by the vertices of the pentagon.

437 Easy

437 of 555

Application +4 | -1

The orthocenter of the triangle with vertices

O\left( {0,\;0} \right),\;A\left( {0,\frac{3}{2}} \right),\;B\left( { - 5,\;0} \right)

View answer and explanation

Correct option: D

\left( {0,\;0} \right)

Option D ((0, 0)) is correct because the orthocenter of a right-angled triangle is the vertex at the right angle, which is the origin in this case.

438 Easy

438 of 555

Application +4 | -1

Through the point

(13,31)

a straight line is drawn to meet the axes of

X

and

Y

Q

and

S

respectively. If the rectangle

OQRS

is completed, the co-ordinates of

{R}

satisfy the equation.

View answer and explanation

Correct option: A

\frac{13}{x}+\frac{31}{y}=1

Option A (

\frac{13}{x}

\frac{31}{y}

=1) is correct because it represents the equation satisfied by the coordinates of point R, derived from the given geometric conditions.

439 Easy

439 of 555

Analyzing +4 | -1

Two vertices of a triangle are

(3,-2)

and

(-2,3)

and its orthocentre is

(-6,1)

. Then the third vertex of this triangle can not lie on the line

View answer and explanation

Correct option: C

5 x+y=2

Option C (

5 x+y=2

) is correct because the third vertex, calculated using the orthocenter properties, does not lie on this line.

440 Easy

440 of 555

Application +4 | -1

A line is drawn from

P\left( {{x_1},{y_1}} \right)

in the direction

\alpha

with the x-axis, to meet

A\;x + B\;y + C = 0

at Q. Then, the length PQ is equal to

View answer and explanation

Correct option: B

- \frac{{A\;{x_1} + B\;{y_1} + C}}{{A\cos \alpha + B\sin \alpha }}

Option B (

- \frac{{A\;{x_1} + B\;{y_1} + C}}{{A\cos \alpha + B\sin \alpha }}

) is correct because it represents the distance from the point to the line along the given direction.

441 Medium

441 of 555

Application +4 | -1

\cos \theta=\frac{8}{17}

and

\theta

lies between the Ist quadrant, then the value of

\cos \left(30^{\circ}+\theta\right)+\cos \left(45^{\circ}-\theta\right)+\cos \left(120^{\circ}-\theta\right)

View answer and explanation

Correct option: A

\frac{23}{17}\left(\frac{\sqrt{3}-1}{2}+\frac{1}{\sqrt{2}}\right)

Option A (

\frac{{23}}{{17}}(\frac{{\sqrt{{3}}-1}}{{2}}+\frac{{1}}{{\sqrt{{2}}}})

) is correct because it is derived by expanding the trigonometric functions and using the given value of

\cos \theta

442 Medium

442 of 555

Application +4 | -1

A line is such that its segments between the straight lines

5 x-y=4

and

3 x+4 y-4=0

is bisected at the points

(1,5)

. Its equation is

View answer and explanation

Correct option: C

83 x-35 y+92=0

Option C (

83 x-35 y+92=0

) is correct because it represents the line whose segment between the two given lines is bisected at the point (1,5).

443 Easy

443 of 555

Application +4 | -1

The equation of the line perpendicular to the line

2 x+3 y+5=0

and passing through

(1,1)

, is

View answer and explanation

Correct option: C

3 x-2 y-1=0

Option C (

3 x-2 y-1=0

) is correct because it represents the equation of a line perpendicular to the given line and passing through the specified point.

444 Easy

444 of 555

Application +4 | -1

The line passing through the point of intersection of

x + y = 2,\;x - y = 0

and is parallel to

x + 2y = 5,

View answer and explanation

Correct option: D

x + 2y = 3

Option D (

x + 2y = 3

) is correct because it represents the line passing through the intersection of the given lines and is parallel to

x + 2y = 5

445 Easy

445 of 555

Application +4 | -1

The vertices of a triangle are (6, 0), (0, 6) and (6, 6). The distance between its circumcentre and centroid is

View answer and explanation

Correct option: B

\sqrt 2

Option B (

\sqrt 2

) is correct because the distance between the circumcentre and centroid of the triangle is calculated to be

\sqrt 2

446 Easy

446 of 555

Application +4 | -1

The given line

3 x-4 y=8

, the points

(2,3),(-4,5)

are on the

View answer and explanation

Correct option: A

same sides

Option A (same sides) is correct because when the points are substituted into the line's equation, the resulting values have the same sign.

447 Medium

447 of 555

Application +4 | -1

A(4,4), B(3,5)

and

C(-1,-1)

are the vertices of

View answer and explanation

Correct option: B

a right angled triangle

Option B (a right angled triangle) is correct because the slopes of AB and AC are negative reciprocals, indicating that AB is perpendicular to AC, thus forming a right angle at A.

448 Easy

448 of 555

Application +4 | -1

Three points are

A\left( {6,3} \right),B\left( { - 3,5} \right),C\left( {4, - 2} \right)

and

P\left( {x,y} \right)

is a point, then the ratio of area of

\Delta PBC

and

\Delta ABC

View answer and explanation

Correct option: A

\left| {\frac{{x + y - 2}}{7}} \right|

Option A is correct because the ratio of areas is found by dividing the area of triangle PBC by the area of triangle ABC, using the determinant formula for the area of a triangle.

449 Easy

449 of 555

Application +4 | -1

A(2,3)

and

B(-2,1)

are two vertices of

a

triangle and third vertex moves on the line

2 x+3 y=9

, then the locus of the centroid of the triangle is

View answer and explanation

Correct option: A

2 x+3 y=1

The correct answer is A (

2 x+3 y=1

) because it represents the locus of the centroid of the triangle, derived by substituting the coordinates of the third vertex on the given line into the centroid formula.

450 Easy

450 of 555

Application +4 | -1

Reduce the equation

\sqrt{3} x+y-8=0

into normal form. The value of

p

View answer and explanation

Correct option: A

Option A (4) is correct because after normalizing the equation, the constant term on the right-hand side represents the perpendicular distance (p) from the origin.

Revision notes and key takeaways

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