Class 9 Advanced Maths Chapter 4 Coordinate Geometry Ex 4.4 Solutions - #NCSOLVE 📚

Accurate and updated NCERT Class 9 Advanced Maths Solutions Chapter 4 Coordinate Geometry Ex 4.4 help students build a strong mathematical foundation.

Ex 4.4 Class 9 Advanced Maths Solutions

Advanced Maths Class 9 Exercise 4.4 Solutions

Exercise 4.4 Class 9 Advanced Maths Solutions

Question 1.
The line kx + 3y – 12 = 0 forms a right angled triangle with the x and y-coordinate axes. If the total area of this triangle is 12 sq units, find all possible values for the slope of this line.
Solution:
Given, the equation of the line is kx + 3y – 12 = 0.
To find the intercepts, we put y = 0 for the x-intercept and x = 0 for the y-intercept.
For x-intercept,
kx – 12 = 0 ⇒ x = \(\frac{12}{k}\)
∴ The coordinates of the vertex on the X-axis are A(\(\frac{12}{k}\), 0)
For y-intercept,
3y – 12 = 0 ⇒ y = 4
∴ The coordinates of the vertex on the Y-axis are B(0, 4).
The line forms a right angled triangle with the origin O(0, 0) and the axes.
The lengths of the base and height of this triangle are \(\frac{12}{k}\) and 4, respectively.
We know that the area of a triangle = \(\frac{1}{2}\) × Base × Height
Given, area of the triangle = 12 sq units
⇒ \(\frac{1}{2}\) × |\(\frac{12}{k}\)| × |4| = 12
⇒ \(\frac{24}{k}\) = 12
⇒ |k| = 2
⇒ k = 2 or k = -2
Now, the slope (m) of the line kx + 3y – 12 = 0 is given by m = \(-\frac{\text { Coefficient of } x}{\text { Coefficient of } y}\)
⇒ m = –\(\frac{k}{3}\)
Case I If k = 2, m = –\(\frac{2}{3}\)
Case II If k = -2, m = \(\frac{(-2)}{3}\) = \(\frac{2}{3}\)
Hence, the possible values for the slope of the line are \(\frac{2}{3}\) and – \(\frac{2}{3}\)

Question 2.
The straight line px + qy + r = 0 (where p, q, and r ≠ 0) forms an isosceles right angled triangle with the coordinate axes in the quadrant. What must be the algebraic relationship between the coefficients p and q?
Solution:
Given, the equation of the line is px + qy + r = 0, where p, q, r ≠ 0.
To find the intercepts made by the line on the coordinate axes, we put y = 0 for the x-intercept and x = 0 for the y-intercept.
For x-intercept
px + r = 0 ⇒ x = –\(\frac{r}{p}\)
∴ The vertex of the triangle on the X-axis is A (-\(\frac{r}{p}\), 0).
For y-intercept
qy + r = 0 ⇒ y = –\(\frac{r}{p}\),
∴ The vertex of the triangle on the X-axis is B (0, \(\frac{r}{q}\)).
The triangle is formed by these intercepts and the origin O(0, 0).
We know that a triangle formed by the coordinate axes is always right angled at the origin.
Also, given the triangle is an isosceles right angled triangle.
∴ Length of base OA = Length of height OB r 9

Hence, the algebraic relationship between the coefficients p and q must be |p| = |q|.

Question 3.
Line l₁ has the equation 3x – 5y + 10 = 0.
Line l₂ has the equation 5x + 3y + K = 0.
Prove algebraically that l₁, and l₂ are perpendicular.
Also, if the x-intercept of l₁ is identical to the y-intercept of l₂ then find the value of K.
Solution:
Given, the equation of line l₁, is 3x – 5y + 10 = 0
and the equation of line l₂ is 5x + 3y + K = 0.
We know that the slope m of a line Ax + By + C- 0 is
given by m = –\(\frac{A}{B}\).
Slope of line l₁, m₁ = –\(\frac{3}{-5}\) = \(\frac{3}{5}\)
and slope of line l₂, m₂ = –\(\frac{5}{3}\).
Now, m₁ × m₂ = (\(\frac{3}{5}\)) –\(\frac{5}{3}\) = -1
Since, the product of the slopes is -1, the lines l₁ and l₂ are perpendicular.
For x-intercept of l₁ we put y = 0 in its equation.
∴ 3x – 5(0) + 10 = 0 ⇒ 3x = -10 ⇒ x = –\(\frac{10}{3}\)
For y-intercept of l₂, we put x = 0 in its equation.
∴ 5(0) + 3y + K = 0
⇒ 3y = -K ⇒ y = –\(\frac{K}{3}\)
Also, given the x-intercept of l₁ is identical to the
y-intercept of l₂ i.e. –\(\frac{10}{3}\) = –\(\frac{K}{3}\) ⇒ K = 10.

Question 4.
A line is written in the general form kx – y + C = 0.
You are given two clues about this line.
Clue 1 : The line passes through the coordinate point (3, 10).
Clue 2 : The sum of its x-intercept and y-intercept is exactly equal to its slope. Find all possible equations for this line.
Solution:
Given, the general form of the line is kx – y + C = 0.
For x-intercept, put y = 0 in given equation of line.
∴ kx – 0 + C = 0 ⇒ x = –\(\frac{C}{k}\)
For y-intercept, put x = 0 in given equation of line
∴ k(0) – y + C = 0
⇒ y = C
Now, slope of given line = –\(\frac{k}{(-1)}\) = k
Since, the given line passes through the coordinate point (3, 10).
∴ k(3) – 10 + C = 0
⇒ 3k + C = 10
⇒ C = 10 – 3k ……….. (i)
Also, the sum of x-intercept and y-intercept of given line is equal to its slope.
∴ –\(\frac{C}{k}\) + C = k
⇒ \(\frac{-C+C k}{k}\) = k ⇒ C(k – 1) = k²
On substituting the value of C from Eq. (i) into Eq.(ii), we get
(10 – 3k) (k – 1) = k²
⇒ 10k – 10 – 3k² + 3k = k²
⇒ -3k² + 13k – 10 = k²
⇒ 4k² – 13k + 10 = 0
⇒ 4k² – 8k – 5k + 10 = 0
⇒ 4k(k – 2) – 5(k – 2) = 0
⇒ (4k – 5)(k – 2) = 0
∴ k = 2 or k = \(\frac{5}{4}\)
Case I When k = 2, C = 10 – 3 (2) = 4 [from Eq. (i)]
∴ The equation of required line is 2x – y + 4 = 0.
∴ Case II When k = \(\frac{5}{4}\),
C = 10 – 3(\(\frac{5}{4}\)) = 10 – \(\frac{15}{4}\) = \(\frac{25}{4}\) [from Eq. (i)]
∴ The equations of required line is
\(\frac{5}{4}\) x – y + \(\frac{25}{4}\) = 0i.e. 5x – 4y + 25 = 0.

The post Class 9 Advanced Maths Chapter 4 Coordinate Geometry Ex 4.4 Solutions appeared first on Learn CBSE.

📚 NCsolve - Your Global Education Partner 🌍

Empowering Students with AI-Driven Learning Solutions

Welcome to NCsolve — your trusted educational platform designed to support students worldwide. Whether you're preparing for Class 10, Class 11, or Class 12, NCsolve offers a wide range of learning resources powered by AI Education.

Our platform is committed to providing detailed solutions, effective study techniques, and reliable content to help you achieve academic success. With our AI-driven tools, you can now access personalized study guides, practice tests, and interactive learning experiences from anywhere in the world.

🔎 Why Choose NCsolve?

At NCsolve, we believe in smart learning. Our platform offers:

• ✅ AI-powered solutions for faster and accurate learning.
• ✅ Step-by-step NCERT Solutions for all subjects.
• ✅ Access to Sample Papers and Previous Year Questions.
• ✅ Detailed explanations to strengthen your concepts.
• ✅ Regular updates on exams, syllabus changes, and study tips.
• ✅ Support for students worldwide with multi-language content.

🌐 Explore Our Websites:

🔹 ncsolve.blogspot.com
🔹 ncsolve-global.blogspot.com
🔹 edu-ai.blogspot.com

📲 Connect With Us:

👍 Facebook: NCsolve
📧 Email: ncsolve@yopmail.com

#NCsolve #EducationForAll #AIeducation #WorldWideLearning #Class10 #Class11 #Class12 #BoardExams #StudySmart #CBSE #ICSE #SamplePapers #NCERTSolutions #ExamTips #SuccessWithNCsolve #GlobalEducation