A variable line, drawn through the point of intersection of the straight lines x a + y b = 1 and x b + y a = 1 , meets the coordinate axes in A & B . Show that the locus of the mid point of A B is the curve 2 x y ( a + b ) = a b ( x + y ) .

The given equations of the lines are x a + y b = 1 and x b + y a = 1 ⇒ b x + a y - a b = 0 and a x + b y - a b = 0 . The equation of a line passing through the point of intersection of the lines L 1 = 0 and L 2 = 0 is L 1 + λ L 2 = 0 . Hence, the equation of a line passing through the point of intersection of the lines b x + a y - a b = 0 & a x + b y - a b = 0 is b x + a y - a b + λ a x + b y - a b = 0 ⇒ x b + λ a + y a + λ b - a b 1 + λ = 0 . To find the x -intercept of the line, put y = 0 , to get x b + λ a - a b 1 + λ = 0 ⇒ x = a b 1 + λ b + λ a . To find the y -intercept of the line, put x = 0 , to get y a + λ b - a b 1 + λ = 0 ⇒ y = a b 1 + λ a + λ b . Thus, the points on the axes are respectively A a b 1 + λ b + λ a , 0 and B 0 , a b 1 + λ a + λ b . The mid-point of the line segment joining the points x 1 , y 1 and x 2 , y 2 is x 1 + x 2 2 , y 1 + y 2 2 . Hence, the mid-point of the line segment joining A and B is h , k = a b 1 + λ 2 b + λ a , a b 1 + λ 2 a + λ b ⇒ h = a b 1 + λ 2 b + λ a , k = a b 1 + λ 2 a + λ b Now, to obtain the locus, we have to eliminate λ , From the above relation, we have 2 h b + λ a = a b 1 + λ and 2 k a + λ b = a b 1 + λ ⇒ 2 h b + 2 h λ a = a b + a b λ and 2 k a + 2 k λ b = a b + a b λ ⇒ λ 2 a h - a b = a b - 2 h b and λ 2 k b - a b = a b - 2 k a ⇒ λ = b a - 2 h a 2 h - b and λ = a b - 2 k b 2 k - a On comparing, we get b a - 2 h a 2 h - b = a b - 2 k b 2 k - a ⇒ b 2 a - 2 h 2 k - a = a 2 b - 2 k 2 h - b ⇒ 2 a b 2 k - a 2 b 2 - 4 b 2 k h + 2 a b 2 h = 2 a 2 b h - a 2 b 2 - 4 a 2 h k + 2 a 2 b k ⇒ 2 a b 2 k + 2 a b 2 h - 2 a 2 b h - 2 a 2 b k = 4 b 2 h k - 4 a 2 h k ⇒ 2 a b 2 k + h - 2 a 2 b h + k = 4 h k b 2 - a 2 ⇒ 2 a b k + h b - a = 4 h k b - a b + a ⇒ a b h + k = 2 h k a + b Now, to get the locus replace h , k by x , y , to get a b x + y = 2 x y a + b .

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Locus of the image of the point $(2, 3)$ in the line $(2 x - 3 y + 4) + k (x - 2 y + 3) = 0, k \in R,$ is a

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Important Questions on Point and Straight Line

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-3>Q 28

Let the orthocentre and centroid of a triangle be

A (- 3, 5)

and

B (3, 3)

. respectively. If

C

is the circumcentre of this triangle, then the radius of the circle having line segment

A C

as diameter, is

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-3>Q 29

If a circle of radius

R

passes through the origin

O

and intersects the coordinate axes at

A

and

B

, then the locus of the foot of perpendicular from

O

A B

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-3>Q 30

Let

(x, y)

be any point on the parabola

y^{2} = 4 x .

Let

P

be the point that divides the line segment from

(0, 0)

(x, y)

in the ratio

1 : 3 .

Then the locus of

P

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-4>Q 1

The vertices of a triangle

O B C

are

O (0, 0) B (- 3, - 1)

and

C (- 1, - 3) .

Find the equation of line parallel to

B C

and intersecting the sides

O B

and

O C,

whose perpendicular distance from the point

(0, 0)

\frac{1}{2}

units.

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-4>Q 2

A variable line, drawn through the point of intersection of the straight lines

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

and

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

meets the coordinate axes in

A & B .

Show that the locus of the mid point of

A B

is the curve

2 x y (a + b) = a b (x + y) .

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-4>Q 3

From the vertices

A, B, C

of a triangle

A B C

perpendiculars

A D, B E, C F

are drawn to any straight line. Show that the perpendiculars from

D, E, F

B C, C A, A B

respectively are concurrent.

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-4>Q 4

A triangle is formed by the lines whose equations are

A B : x + y - 5 = 0, B C : x + 7 y - 7 = 0

and

C A : 7 x + y + 14 = 0 .

Find the bisector of the interior angle at

B

and the exterior angle at

C

. Determine the nature of the interior angle at

A

and find the equation of the bisector.

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-4>Q 5

Let in

∆ P A B, A

(0, 0), B

(a, 0)

and

P

is a variable point such that

∠ P B A

is equal to three times

∠ P A B

. Then find the locus of

P .

Locus of the image of the point 2, 3 in the line 2x-3y+4+kx-2y+3=0, k∈R, is a

Important Questions on Point and Straight Line

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Locus of the image of the point $(2, 3)$ in the line $(2 x - 3 y + 4) + k (x - 2 y + 3) = 0, k \in R,$ is a