RD Sharma Solutions Class 12 The Plane Exercise 29.15

RD Sharma Solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.15, is provided here for students to prepare for their exams at ease. Students can refer to RD Sharma Solutions for Class 12, which is the best reference material developed by our expert tutors, to help students gain knowledge and to have a good grip over the subject. Solving these exercises will ensure students obtain a good score in their exams. The pdf consisting of this chapter solutions which are available for free can be downloaded from the links given below.

Download PDF of RD Sharma Solutions For Class 12 Maths Chapter 29 Exercise 15

 

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Access RD Sharma Solutions For Class 12 Maths Chapter 29 Exercise 15

EXERCISE 29.15

Q1.

Solution:

Given:

Let point P = (0, 0, 0) and M be the image of P in the plane

3x + 4y – 6z + 1 = 0.

Direction ratios of PM are proportional to 3, 4, – 6 as PM is normal to the plane.

The equation of the line passing through (x1, y1, z1) and having direction ratios proportional to l, m, n is given by

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 157

Let M = (3α, 4α, –6α).

As M is the image of P in the given plane, the midpoint of PM lies on the plane.

Using the midpoint formula, we have

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 158

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 159

Q2.

Solution:

Given:

Let point P = (1, 2, –1) and M be the image of P in the plane 3x – 5y + 4z = 5.


Direction ratios of PM are proportional to 3, –5, 4 as PM is normal to the plane.

The equation of the line passing through (x1, y1, z1) and having direction ratios proportional to l, m, n is given by

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 160

Let M = (3α + 1, –5α + 2, 4α – 1).

As M is the image of P in the given plane, the midpoint of PM lies on the plane.

Using the midpoint formula, we have

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 161

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 162

Q3.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 163

So, Q = (2α – 1, 3α + 3, –α + 1)


Now, let us find the direction ratios of PQ.

The direction ratios of a line joining two points (x1, y1, z1) and (x2, y2, z2) are given by (x2 – x1, y2 – y1, z2 – z1).

Here,

(x1, y1, z1) = (5, 4, 2) and (x2, y2, z2) = (2α – 1, 3α + 3, –α + 1)


Direction Ratios of PQ are ((2α – 1) – (5), (3α + 3) – (4), (–α + 1) – (2))

Direction Ratios of PQ are (2α – 6, 3α – 1, –α – 1)

PQ is perpendicular to the given line, whose direction ratios are (2, 3, –1).


We know that if two lines with direction ratios (a1, b1, c1) and (a2, b2, c2) are perpendicular to each other, then a1a2 + b1b2 + c1c2 = 0.

(2)(2α – 6) + (3)(3α – 1) + (–1)(–α – 1) = 0

4α – 12 + 9α – 3 + α + 1 = 0

14α – 14 = 0

14α = 14

α = 1


We have Q = (2α – 1, 3α + 3, –α + 1)

Q = (2×1 – 1, 3×1 + 3, –1 + 1)

Q = (1, 6, 0)


Using the distance formula, we have

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 164

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 165

Q4.

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 166

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 167

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 168
RD Sharma Solutions for Class 12 Maths Chapter 29 - image 169

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 170

Q5.

Solution:

Given:

Let point P = (1, 1, 2) and Q be the foot of the perpendicular drawn from P to the plane 2x – 2y + 4z + 5 = 0.


Direction ratios of PQ are proportional to 2, –2, 4 as PQ is normal to the plane.


The equation of the line passing through (x1, y1, z1) and having direction ratios proportional to l, m, n is given by

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 171

This point lies on the given plane, which means this point satisfies the plane equation.

2(2α + 1) – 2(–2α + 1) + 4(4α + 2) + 5 = 0

4α + 2 + 4α – 2 + 16α + 8 + 5 = 0

24α + 13 = 0

24α = –13

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 172

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 173

RD Sharma Solutions for Class 12 Maths Chapter 29 - image 174

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