Решения NCERT класса 11 — Глава 10 Прямые линии — Упражнение 10.3 | Набор 2

Вопрос 11. Докажите, что прямая, проходящая через точку (x1, y1) и параллельная прямой Ax + By + C = 0, равна A (x – x1) + B (y – y1) = 0.

Решение:

Let’s assume that the slope of line Ax + By + C = 0 be m,

Ax + By + C = 0

Therefore, y = -A/B x – C/B

m = -A / B

By using the formula,

As we know that the equation of the line passing through point (x1, y1) and having slope m = -A/B is,

y – y1 = m (x – x1)

y – y1= -A/B (x – x1)

B (y – y1) = -A (x – x1)

Hence, A(x – x1) + B(y – y1) = 0

Therefore, the line through point (x1, y1) and parallel to the line Ax + By + C = 0 is A (x – x1) + B (y – y1) = 0

Hence, proved.

Вопрос 12. Две прямые, проходящие через точку (2, 3), пересекают друг друга под углом 60 ^o . Если наклон одной прямой равен 2, найдите уравнение другой прямой.

Решение:

Given that, m1 = 2

Let’s assume that the slope of the first line be m1 and

The slope of the other line be m2.

Angle between the two lines is 60° (Given)

Therefore,

tanθ = |m1 – m2| / |1 + m1m2|

tan 60^o = |2 – m2| / |1 + 2m2|

√3 = ±((2 – m2) / (1 + 2m2))

After rationalization, we got,

m2 = (2 – √3) / (2√3 + 1) and m2 = -(2 – √3) / (2√3 + 1)

Case 1: When m2 = (2 – √3) / (2√3 + 1)

Therefore, the equation of line passing through point (2, 3) and having slope m2 = (2 – √3) / (2√3 + 1) is :

y – 2 = ((2 – √3) / (2√3 + 1)) x (x – 2)

After solving above equation we got,

(√3 – 2)x + (2√3 + 1) y = 8√3 – 1

Equation of line is (√3 – 2)x + (2√3 + 1) y = 8√3 – 1.

Case 2: When m2 = -(2 – √3) / (2√3 + 1)

Therefore, the equation of line passing through point (2, 3) and having slope m2 = -(2 – √3) / (2√3 + 1) is :

y – 3 = -(2 – √3) / (2√3 + 1) x (x – 2)

After solving above equation we got,

(√3 + 2)x + (2√3 – 1) y = 8√3 + 1

Equation of line is (√3 + 2)x + (2√3 – 1) y = 8√3 + 1.

Вопрос 13. Найдите уравнение правой биссектрисы отрезка, соединяющего точки (3, 4) и (–1, 2).

Решение:

Given that,

The right bisector of a line segment bisects the line segment at 90° and

End-points of the line segment AB are given as A (3, 4) and B (–1, 2).

Let’s assume that the mid-point of AB be (x, y)

x = (3-1)/2 = 2/2 = 1

y = (4+2)/2 = 6/2 = 3

(x, y) = (1, 3)

Let’s the slope of line AB be m1

m1 = (2 – 4)/(-1 – 3)

= -2/(-4) = 1/2

Let’s the slope of the line perpendicular to AB be m2

m2 = -1/(1/2) = -2

The equation of the line passing through (1, 3) and having a slope of –2 is,

(y – 3) = -2 (x – 1)

y – 3 = – 2x + 2

2x + y = 5

Hence, the required equation of the line is 2x + y = 5

Вопрос 14. Найдите координаты основания перпендикуляра из точки (–1, 3) к прямой 3x – 4y – 16 = 0.

Решение:

Let us consider the co-ordinates of the foot of the perpendicular from (-1, 3) to the line 3x – 4y – 16 = 0 be (a, b)

Therefore, let the slope of the line joining (-1, 3) and (a, b) be m1

m1 = (b-3)/(a+1) ,

and let the slope of the line 3x – 4y – 16 = 0 be m2

y = 3/4x – 4

m2 = 3/4

Since these two lines are perpendicular, m1 × m2 = -1 (Given)

(b-3) / (a+1) × (3/4) = -1

(3b-9) / (4a+4) = -1

3b – 9 = -4a – 4

4a + 3b = 5 ————(i)

Point (a, b) lies on the line 3x – 4y = 16

3a – 4b = 16 ———-(ii)

after solving equations (i) and (ii), we get

a = 68/25 and b = -49/25

Hence, the co-ordinates of the foot of perpendicular is (68/25, -49/25)

Вопрос 15. Перпендикуляр из начала координат к прямой y = mx + c пересекает ее в точке (–1, 2). Найдите значения m и c.

Решение:

Given that,

The perpendicular from the origin meets the given line at (–1, 2).

As we know that the equation of line is y = mx + c

The line joining the points (0, 0) and (–1, 2) is perpendicular to the given line.

therefore, the slope of the line joining (0, 0) and (–1, 2) = 2/(-1) = -2

Slope of the given line is m. (Assumption)

m × (-2) = -1

m = 1/2

Since, point (-1, 2) lies on the given line,

y = mx + c

2 = 1/2 × (-1) + c

c = 2 + 1/2 = 5/2

Hence, the values of m and c are 1/2 and 5/2 respectively.

Вопрос 16. Если p и q — длины перпендикуляров из начала координат к прямым x cos θ − y sin θ = k cos 2θ и x sec θ + y cosec θ = k соответственно, докажите, что p ² + 4q ² = к ²

Решение:

Given that,

The equations of given lines are

x cos θ – y sin θ = k cos 2θ ———–(i)

x sec θ + y cosec θ = k —————(ii)

Perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by

d = |Ax1 + By1 + C| / √A² + B²

After comparing equation (i) and (ii) we get,

A = cos θ, B = -sin θ and C = -k cos 2θ

Given that p is length of perpendicular from (0, 0) to line (i)

p = |A × 0 + B × 0 + C| / √A² + B²

= |-k cos 2θ| / √cos² θ + sin² θ = k cos 2θ

p = k cos 2θ

Let’s square on both side, and we get,

p² = k² cos² 2θ ————(iii)

Now compare eqn. (ii) with general equation of line i.e. Ax + By + C = 0, and we get,

A = sec θ, B = cosec θ and C = -k

As we know that q is length of perpendicular from (0, 0) to line (ii)

q = |A x 0 + B x 0 + C| / √A² + B² = |C| / √A² + B²

= |-k| / √sec² θ + cosec² θ = k cos θ sin θ

q = k cos θ sin θ

Multiply both sides by 2, we get

2q = 2k cos θ sin θ = k × 2sin θ cos θ

2q = k sin 2θ

Squaring both sides, we get

4q² = k² sin²2θ ————–(iv)

Now add (iii) and (iv) we get

p² + 4q² = k² cos² 2θ + k² sin² 2θ

p² + 4q² = k² (cos² 2θ + sin² 2θ) (As we know that cos² 2θ + sin² 2θ = 1)

Hence, p² + 4q² = k²

Hence, proved.

Вопрос 17. В треугольнике ABC с вершинами A (2, 3), B (4, –1) и C (1, 2) найти уравнение и длину высоты из вершины A.

Решение:

Let’s assume that AD be the altitude of triangle ABC from vertex A.

Therefore, AD is perpendicular to BC

Given that,

Vertices A (2, 3), B (4, –1) and C (1, 2)

Let’s slope of line BC = m1

m1 = (-1 – 2) / (4 – 1)

m1 = -1

Let’s slope of line AD be m2

AD is perpendicular to BC

m1 × m2 = -1

-1 × m2 = -1

m2 = 1

The equation of the line passing through point (2, 3) and having a slope of 1 is,

y – 3 = 1 × (x – 2)

y – 3 = x – 2

y – x = 1

Equation of the altitude from vertex A = y – x = 1

Length of AD = Length of the perpendicular from A (2, 3) to BC

Equation of BC is

y + 1 = -1 × (x – 4)

y + 1 = -x + 4

x + y – 3 = 0 ————-(i)

Perpendicular distance d of a line Ax + By + C = 0 from a point (x1, y1) is given by d = |Ax1 + By1 + C| / √A² + B²

Now compare equation (i) to the general equation of line i.e., Ax + By + C = 0, and we get,

Length of AD = |1 × 2 + 1 × 3 – 3| / √1² + 1² = √2 units (A = 1, B = 1 and C = -3)

Hence, the equation and the length of the altitude from vertex A are y – x = 1 and √2 units respectively.

Вопрос 18. Если p — длина перпендикуляра из начала координат к прямой, пересечения которой на осях равны a и b, то покажите, что 1/p ² = 1/a ² + 1/b ²

Решение:

Equation of a line whose intercepts on the axes are a and b is x/a + y/b = 1

bx + ay = ab

bx + ay – ab = 0 ————-(i)

Perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by

d = |Ax1 + By1 + C| / √A² + B²

After comparing eqn. (i) with general equation of line i.e. Ax + By + C = 0 we get,

A = b, B = a and C = -ab

Let’s assume that if p is length of perpendicular from point (x1, y1) = (0, 0) to line (i), we get

p = |A x 0 + B x 0 – ab| / √a² + b² = |-ab| / √a² + b²

Now square on both the sides we get

p² = (-ab)² / a²+ b²

1 / p² = (a² + b²) / a²b²

Hence, 1/p² = 1/a² + 1/b²

Hence, proved.