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 · Easy

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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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301 Easy

301 of 408

Application +4 | -1

The equation of the line passing through the points

P(2,3)

and

Q(4,5)

View answer and explanation

Correct option: D

x-y+1=0

Option D (

x-y+1=0

) is correct because the equation of a line passing through two points

(x_1, y_1)

and

(x_2, y_2)

can be found using the determinant form or the two-point form, and substituting the given points yields the correct equation.

302 Easy

302 of 408

Analyzing +4 | -1

Let

ABC

be a triangle, two of whose vertices are (15, 0) and (0, 10). If the orthocenter is (6, 9), then the third vertex is

View answer and explanation

Correct option: C

(0, 0)

Option C (0, 0) is correct because the third vertex satisfies the condition derived from the fact that the line passing through A and B is perpendicular to the line through C and the orthocenter O.

303 Easy

303 of 408

Application +4 | -1

The area of the parallelogram formed by the lines

2 x-3 y+a=0, \quad 3 x-2 y-a=0,2 x-3 y+3 a=0

and

3 x-2 y-2 a=0

in square units, is

View answer and explanation

Correct option: B

\frac{2 a^2}{5}

Option B (

\frac{2 a^2}{5}

) is correct because it is the area of the parallelogram calculated using the formula involving the constants and determinant of coefficients.

304 Easy

304 of 408

Analyzing +4 | -1

The number of points on the line

x + y = 4

which are unit distance apart from the line

2x + 2y = 5

View answer and explanation

Correct option: A

Option A (0) is correct because the distance between the two parallel lines is greater than 1, meaning no point on the first line is unit distance from the second.

305 Easy

305 of 408

Application +4 | -1

Lines

2x + y = 1

and

2x + y = 7

are

View answer and explanation

Correct option: A

On the same side of a point

\left( {0,\frac{1}{2}} \right)

The correct answer is A (On the same side of a point (0,1/2)) because when the point (0, 1/2) is substituted into both line equations, the resulting values have the same sign.

306 Easy

306 of 408

Application +4 | -1

Mirror image of

(-3,5)

in line mirror

x-y+2=0

View answer and explanation

Correct option: C

(3, -1)

Option C, (3, -1), is the correct mirror image of the point (-3, 5) with respect to the line x - y + 2 = 0.

307 Easy

307 of 408

Application +4 | -1

The area of the parallelogram formed by the lines

x \cos \alpha+y \sin \alpha=p, x \cos \alpha+y \sin \alpha=q

x \cos \beta+y \sin \beta=r

and

x \cos \beta+y \sin \beta=s

, is

View answer and explanation

Correct option: C

\left|\frac{(p-q)(r-s)}{\sin (\alpha-\beta)}\right|

Option C is correct. The area of the parallelogram formed by the given lines is found using the formula involving the determinant of the coefficients and the difference of the constants, resulting in

\left|\frac{{(p-q)(r-s)}}{{\sin (\alpha -\beta )}}\right|

308 Easy

308 of 408

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.

309 Easy

309 of 408

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.

310 Easy

310 of 408

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.

311 Easy

311 of 408

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.

312 Easy

312 of 408

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.

313 Easy

313 of 408

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.

314 Easy

314 of 408

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.

315 Easy

315 of 408

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.

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316 Easy

316 of 408

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.

317 Easy

317 of 408

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.

318 Easy

318 of 408

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

319 Easy

319 of 408

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.

320 Easy

320 of 408

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.

321 Easy

321 of 408

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.

322 Easy

322 of 408

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.

323 Easy

323 of 408

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.

324 Easy

324 of 408

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.

325 Easy

325 of 408

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.

326 Easy

326 of 408

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.

327 Easy

327 of 408

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

328 Easy

328 of 408

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

329 Easy

329 of 408

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.

330 Easy

330 of 408

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.

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