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A triangle has the lines y = m₁x and y = m₂x as two of its sides, with m₁ and m₂ being roots of the equation bx² + 2hx + a = 0. If H(a, b) is the orthocenter of the triangle, if the equation of the third side is (a + b) (ax + by) = ab(a + b – λh). Find λ

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Important Questions on Straight Lines

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

A straight line through origin

O

meets the lines

3 y = 10 - 4 x

and

8 x + 6 y + 5 = 0

at points

A

and

B

respectively. Then,

O

divides the segment

A B

in the ratio

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

The equation of perpendicular bisectors of sides

A B

and

A C

of a

∆ A B C

are

x - y + 5 = 0

and

x + 2 y = 0

respectively. If the coordinates of vertex

A

are

(1, - 2),

then equation of

B C

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

Two sides of a rhombus are along the lines,

x - y + 1 = 0

and

7 x - y - 5 = 0

. If its diagonals intersect at

(- 1, - 2)

, then which one of the following is a vertex of this rhombus ?

HARD

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

A straight line

L

at a distance of

4

units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of

60 °

with the line

x + y = 0 .

Then an equation of the line

L

is:
Note: In actual JEE Main paper, two options were correct for this question. Hence, we have changed one option.

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

If a straight line passing through the point

P (- 3, 4)

is such that its intercepted portion between the coordinate axes is bisected at

P

, then its equation is :

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

Suppose that the points

(h, k), (1, 2)

and

(- 3, 4)

lie on the line

L_{1} .

If a line

L_{2}

passing through the points

(h, k)

and

(4, 3)

is perpendicular to

L_{1},

then

\frac{k}{h}

equals:

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

A line has slope

m

and

y

-intercept 4. The distance between the origin and the line is equal to

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

If we reduce

3 x + 3 y + 7 = 0

to the form

x \cos α + y \sin α = p

, then the value of

p

HARD

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

O (0, 0), A (1, 2), B (3, 4)

are the vertices of

∆ O A B .

The joint equation of the altitude and median drawn from

O

HARD

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

A square, of each side

2

, lies above the

x

-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle

30 °

with the positive direction of the

x

-axis , then the sum of the

x

-coordinates of the vertices of the square is

:

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

A line cuts off equal intercepts on the co-ordinate axes. The angle made by this line with the positive direction of X-axis is

HARD

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

Let

P S

be the median of the triangle with vertices

P (2, 2), Q (6, - 1)

and

R (7, 3)

. The equation of the line passing through

(1, - 1)

and parallel to

P S

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

The equation of the line passing through the point

(- 3, 7)

with slope zero is

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

A rectangle is inscribed in a circle with a diameter lying along the line

3 y = x + 7 .

If the two adjacent vertices of the rectangle are

(- 8, 5)

and

(6, 5),

then the area of the rectangle

(

in sq. units

)

is:

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

If the perpendicular bisector of the line segment joining the points

P (1, 4)

and

Q (k, 3)

has

y

-intercept equal to

- 4

, then a value of

k

is;

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

The larger of two angles made with the

X

-axis of a straight line drawn through

(1, 2)

so that it intersects the line

x + y = 4

at a point distant

\sqrt{6} / 3

from the point

(1, 2)

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

Line joining the points

(0, 3)

and

(5, - 2)

is a tangent to the curve

y = \frac{a x}{1 + x}

, then

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

Let

b, d > 0

. The locus of all points

P (r, θ)

for which the line

O P

(where

O

is the origin) cuts the line

r \sin θ = b

Q

such that

P Q = d

HARD

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

Let

O = (0, 0);

let

A

and

B

be points respectively on

x

-axis and

y

-axis such that

∠ O B A = 60 °

. Let

D

be a point in the first quadrant such that

O A D

is an equilateral triangle. Then the slope of

D B

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

The points

(0, \frac{8}{3}), (1, 3)

and

(82, 30)

A triangle has the lines y = m1x and y = m2x as two of its sides, with m1 and m2 being roots of the equation bx2 + 2hx + a = 0. If H(a, b) is the orthocenter of the triangle, if the equation of the third side is (a + b) (ax + by) = ab(a + b – λh). Find λ

Important Questions on Straight Lines

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A triangle has the lines y = m₁x and y = m₂x as two of its sides, with m₁ and m₂ being roots of the equation bx² + 2hx + a = 0. If H(a, b) is the orthocenter of the triangle, if the equation of the third side is (a + b) (ax + by) = ab(a + b – λh). Find λ