2nd PUC Basic Maths Question Bank Chapter 15 Circles Ex 15.3

Students can Download Basic Maths Exercise 15.3 Questions and Answers, Notes Pdf, 2nd PUC Basic Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Basic Maths Question Bank Chapter 15 Circles Ex 15.3

Part – A

2nd PUC Basic Maths Circles Ex 15.3 One Mark Questions and Answers

1. Show that the circle.

a

Question 1.
x2 + y2 – 8x + 2y + 16 = 0 touches x= axis.
Answer:
Given g = -4, f = 1, & C = 16
2nd PUC Basic Maths Question Bank Chapter 15 Circles Ex 15.3 - 1

Question 2.
x2 + y2 – 4x + 4y + 4 = 0 touches x-axis.
Answer:
Here g = -2, C = 4
Here (-2)2 = 4 ⇒ g2 = C
∴ The circle touches x-axis.

Question 3.
x2 + y2 + 8x + 6y + 9 = 0 touches y = axis.
Answer:
Here f = 3 & C = 9
& f2 = C ⇒ (3)2 = 9
⇒ Circle touches y-axis.

KSEEB Solutions

Question 4.
x2 + y2 – x + 4y + 4 = 0 touches y-axis.
Answer:
Here f= 2 & C = 4
& f2 = C ⇒ (2)2 = 4
⇒ Circle0 touches y-axis

Question 5.
Show that the circle x2 + y2 + 4x – 3y + 4 = 0 touches x- axis.
Answer:
Here g = 2 & C = 4
g2 = C ⇒ (2)2 = 4
⇒ Circle touches x-axis.

Question 6.
Show that the circle x2 + y2 – 3x + 8y + 16 = 0 touches y – axis.
Answer:
Here f = 4, C = 16 & f2 = C
⇒ (4)2 = 16 .
∴ Circle touches y-axis.

Part – B

2nd PUC Basic Maths Circles Ex 15.3 Two Marks Questions and Answers

Question 1.
Find the length of the chord of the circle x2 + y2 – 6x + 4y + 5 = 0 intercepted by x-axis.
Answer:
Centre of given circle = (3,-2), c = 5
Length of chord intercepted by x – axis = 2\(\sqrt{g^{2}-c}\) = 2\(\sqrt{9-5}\) = 2\(\sqrt{4}\) = 2.2 = 4 units.

Question 2.
Find the length of the chord of the circle x2 + y2 – 6x + 15y – 16 = 0 intercepted by the y-axis.
Answer:
centre = \(\left(3, \frac{-15}{2}\right)\) & C = -16
Length of the chord intercepted by y – axis = 2\(\sqrt{f^{2}-c}\) = 2\(\sqrt{\left(\frac{15}{2}\right)^{2}+16}\)
2nd PUC Basic Maths Question Bank Chapter 15 Circles Ex 15.3 - 2

Question 3.
Find the length of the chord of the circle x2 + y2 + 3x – 2 = 0 intercepted by y-axis.
Answer:
Centre = \(\left(-\frac{3}{2}, 0\right)\) c = -2
Length of the chord by y – axis = 2\(\sqrt{f^{2}-c}\) = 2\(\sqrt{0+2}\) = 2\(\sqrt{2}\) units.

KSEEB Solutions

Question 4.
Find the length of the chord of the circle x2 + y2 + 3x – y – 6 = 0 intercepted by the y-axis.
Answer:
Centre = \(\left(-\frac{3}{2}, \frac{1}{2}\right)\) c = -6
Length of the chord by axis = 2 \(\sqrt{f^{2}-c}\) = 2\(\sqrt{\frac{1}{4}+6}\) = 2\(\sqrt{\frac{25}{4}}\) = 5 units

Question 5.
Find the length of the chord of the circle x2 + y2 + 4x + 6y – 12 = 0 and x + 4y – 6 = 0.
Answer:
2nd PUC Basic Maths Question Bank Chapter 15 Circles Ex 15.3 - 3

Question 6.
Find the length of the chord intercepted by the circle x2 + y2 – 8x – 6y – 6 = 0 and the line x – 7 y – 8 = 0.
Answer:
2nd PUC Basic Maths Question Bank Chapter 15 Circles Ex 15.3 - 4

Question 7.
Find the length of the chord intercepted by the circle x2 + y2 – 6x – 2y + 5 = 0 and the line x – y + 1 = 0.
Answer:
2nd PUC Basic Maths Question Bank Chapter 15 Circles Ex 15.3 - 5

Question 8.
Prove that the length of the chord x + 2y = 5 of the circle x2 + y2 = 9 is 4 units.
Answer:
Length of the chord intercepted by x – axis = 2\(\sqrt{g^{2}-c}\)
Here g = 5, f = -3, c = 9
∴ Length = \(2 \cdot \sqrt{25-9}=2 \sqrt{16}\)
= 2(4) = 8 units
Hence proved.
Centre of the given circle = (0,0), r = 3 units line = x + 2y – 5 = 0
\(d=\left|\frac{0+2(0)-5}{\sqrt{1^{2}+2^{2}}}\right|=\frac{5}{\sqrt{5}}=\sqrt{5}\)
∴ Length of the chord = \(2 \cdot \sqrt{r^{2}-d^{2}}\)
\(=2 \cdot \sqrt{(3)^{2}-(\sqrt{5})^{2}}=2 \sqrt{9-5}=2.2=4 \text { units. }\)
Hence proved.

KSEEB Solutions

Part – C

2nd PUC Basic Maths Circles Ex 15.3 Five Marks Questions and Answers

Question 1.
Show that the line 3x + 4y + 7 = 0 touches the circle x2 + y2 – 4x – 6y – 12 = 0 and find the point of contact.
Answer:
Given centre = (2, 3) and r = \(\sqrt{4+9+12}\) = \(\sqrt{25}\) = 5
If the length of the perpendicular is equal to radius of the circle than the line touches the circle
2nd PUC Basic Maths Question Bank Chapter 15 Circles Ex 15.3 - 6
∴ The line 3x + 4y + 7 touches the circle
Any line perpendicular to 3x + 4y + 7 = 0 is of the form 4x – 3y + k = 0
It passes through (2,3) ∴ 8 – 9 + k = 0 ⇒ k = 1
∴ The line is 4x – 3y + 1 = 0
Point of contact is the intersection of 3x + 4y + 7 = 0 and 4x – 3y + 1 = 0
2nd PUC Basic Maths Question Bank Chapter 15 Circles Ex 15.3 - 7
∴ The point of contact is (-1,-1)

Question 2.
Find k if the line 3x + y + k = 0 touches the circle x2 + y2 – 2x – 4y – 5 = 0.
Answer:
Given C = (1, 2) C = -5
Radius = Length of the perpendicular from (1, 2) to 3x + y + k = 0.
2nd PUC Basic Maths Question Bank Chapter 15 Circles Ex 15.3 - 8

KSEEB Solutions

Question 3.
Find k if the 2x + y + k = 0 touches the circle x2 + y2 + 6x + 2y + 5 = 0.
Answer:
Hence centre = (-3,-1), c = 5
\(r=\sqrt{g^{2}+f^{2}+c}=\sqrt{9+1-5}=\sqrt{5}\)
r = length of the perpendicular from (-3,-1) to the line 2x + y + k = 0
\(r=\sqrt{g^{2}+f^{2}+c}=\sqrt{9+1-5}=\sqrt{5}\)
2nd PUC Basic Maths Question Bank Chapter 15 Circles Ex 15.3 - 9

Question 4.
Find k if the line 4x – y + k = 0 touches the circle x2 + y2 + 4x – 8y + 3 = 0.
Answer:
Here centre = (-2,4), r = \(\sqrt{4+16-3}\) = \(\sqrt{17}\)
radius = length of perpendicular from (-2, 4) to the line 4x – y + k = 0
∴ \(\sqrt{17}\) = \(\left|\frac{4(-2)-4+k}{\sqrt{16+1}}\right|\)
-12 + k = ±\((\sqrt{17})^{2}\) ⇒ -12 + K = ±17 ⇒ k = 28 or -5

Question 5.
Find k such that the line x – 2y + k = 0 touches the circle x2 + y2 + 3x – 2y + 2 = 0.
Answer:
Here centre \(\left(-\frac{3}{2}, 1\right)\), \(r=\sqrt{\frac{9}{4}+1-2}=\sqrt{\frac{5}{4}}\)
r = length of perpendicular from \(\left(-\frac{3}{2},-1\right)\) to the line x – 2y + k = 0
2nd PUC Basic Maths Question Bank Chapter 15 Circles Ex 15.3 - 10

a