# Example 22

Last updated at March 9, 2017 by Teachoo

Last updated at March 9, 2017 by Teachoo

Transcript

Example 22 Assuming that straight lines work as the plane mirror for a point, find the image of the point (1, 2) in the line x − 3y + 4 = 0 . Let line AB be x – 3y + 4 = 0 & point P be (1, 2) Let Q (h, k) be the image of point P (1, 2) in line AB Since line AB is mirror Point P & Q are at equal distance from line AB, i.e. PR = QR, i.e. R is the mid point of PQ Image is formed perpendicular to mirror i.e. line PQ is perpendicular to line AB Since R is the mid point of PQ We know that mid point of a line joining (x1, y1) & (x2, y2) = ((𝑥_1+〖 𝑥〗_2)/2, (𝑦_1+ 𝑦_2)/2) Mid point of PQ joining (1, 2) & (h, k) is = ((1 + ℎ)/2, (2 + 𝑘)/2) Coordinate of point R = ((1 + ℎ)/2, (2 + 𝑘)/2) Since point R lies on the line AB It will satisfy the equation of line AB Putting x = (ℎ + 1)/2 & y = (𝑘 + 2)/2 in equation of AB x – 3y + 4 = 0 ((ℎ + 1)/2) – 3((𝑘 + 2)/2) + 4 = 0 (ℎ + 1 − 3(𝑘 + 2) + 4 × 2)/2 = 0 h + 1 – 3k – 6 + 8 = 0 h – 3k + 1 – 6 + 8 = 0 h – 3k + 3 = 0 h – 3k = –3 Also, PQ is perpendicular to AB We know that If two lines are perpendicular then product of their slope is equal to – 1 ∴ Slope of AB × Slope of PQ = –1 Slope of PQ = (−1)/(𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝐴𝐵) Finding slope of AB Equation of line AB is x – 3y + 4 = 0 x + 4 = 3y 3y = x + 4 y = (𝑥 + 4)/3 y = (1/3)x + 4/3 The above equation is of the form y = mx + c where m = Slope of line So, Slope of line AB = 1/3 Now, Slope of line PQ = (−1)/(𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝐴𝐵) = (−1)/(1/3) = –3 Now, Line PQ is formed by joining points P(1, 2) & Q(h, k) Slope of PQ = (𝑦_2 − 𝑦_1)/(𝑥_2− 𝑥_1 ) –3 = (𝑘 − 2)/(ℎ − 1) –3(h – 1) = k – 2 –3h + 3 = k – 2 –3h – k = –2 – 3 –3h – k = –5 –(3h + k) = –5 3h + k = 5 Now, our equations are h – 3k = –3 …(1) & 3h + k = 5 …(2) From (1) h – 3k = –3 h = 3k – 3 Putting value of h in (2) 3h + k = 5 3(3k – 3) + k = 5 9k – 9 + k = 5 9k + k = 5 + 9 10k = 14 k = 14/10 k = 7/5 Putting k = 7/5 in (1) 3h + k = 5 3h + 7/5 = 5 3h = 5 – 7/5 3h = (5(5) − 7)/5 3h = (25 − 7)/5 3h = 18/5 h = 18/(5 × 3) h = 6/5 Hence Q = (6/5, 7/5) Hence, image is (6/5, 7/5)

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Example 22 Important You are here

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Class 11

Important Question for exams Class 11

- Chapter 1 Class 11 Sets
- Chapter 2 Class 11 Relations and Functions
- Chapter 3 Class 11 Trigonometric Functions
- Chapter 4 Class 11 Mathematical Induction
- Chapter 5 Class 11 Complex Numbers
- Chapter 6 Class 11 Linear Inequalities
- Chapter 7 Class 11 Permutations and Combinations
- Chapter 8 Class 11 Binomial Theorem
- Chapter 9 Class 11 Sequences and Series
- Chapter 10 Class 11 Straight Lines
- Chapter 11 Class 11 Conic Sections
- Chapter 12 Class 11 Introduction to Three Dimensional Geometry
- Chapter 13 Class 11 Limits and Derivatives
- Chapter 14 Class 11 Mathematical Reasoning
- Chapter 15 Class 11 Statistics
- Chapter 16 Class 11 Probability

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CA Maninder Singh

CA Maninder Singh is a Chartered Accountant for the past 8 years. He provides courses for Practical Accounts, Taxation and Efiling at teachoo.com .