The centre of a circle is (2a, a – 7). Find the values of a if the circle passes through the point (11, –9) and has diameter 10√2 units

Solution:

Given, the center of a circle is (2a, a - 7)

The circle passes through the point (11, -9)

The diameter of the circle is 10√2 units.

We have to find the value of a.

Distance between the center and the point = radius of the circle

Radius of the circle = 10√2/2 = 5√2 units

The distance between two points P (x₁ , y₁) and Q (x₂ , y₂) is

√[(x₂ - x₁)² + (y₂ - y₁)²]

Distance between (2a, a -7) and (11, -9) = 5√2

√[(11 - 2a)² + (-9 - (a - 7))²] = 5√2

√[(11 - 2a)² + (-9 - a + 7)²] = 5√2

√[(11 - 2a)² + (-a - 2)²] = 5√2

On squaring both sides,

(11- 2a)² + (a + 2)² = 50

By using algebraic identity,

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

(11 - 2a)² = 121 - 44a + 4a²

(a + 2)² = a² + 4a + 4

Now, 121 - 44a + 4a² + a² + 4a + 4 = 50

By grouping,

4a² + a² - 44a + 4a + 121 + 4 = 50

5a² - 40a + 125 - 50 = 0

5a² - 40a + 75 = 0

Dividing by 5,

a² - 8a + 15 = 0

a² - 5a - 3a + 15 = 0

a(a - 5) - 3(a - 5) = 0

(a - 3)(a - 5) = 0

Now, a - 3 = 0

a = 3

Also, a - 5 = 0

a = 5

Therefore, the values of a are 3 and 5.

✦ Try This: If a circle drawn with origin as the centre passes through (13/2 ,0), then the point which does not lie in the interior of the circle is a. (-3/4, 1), b. (2, 7/3, c. (5, -1/2), d. (-6, 5/2)

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 7

NCERT Exemplar Class 10 Maths Exercise 7.3 Problem 14

The centre of a circle is (2a, a – 7). Find the values of a if the circle passes through the point (11, –9) and has diameter 10√2 units

Summary:

The centre of a circle is (2a, a – 7). The values of a if the circle passes through the point (11, –9) and has diameter 10√2 units are 3 and 5

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