Question

If the line 3x + 4y = m touches the circle x 2 + y2 = 10x, then m is equal to

Answer 1

If the line 3x + 4y = m touches the circle x 2 + y2 = 10x, then m is equal to (40, -10)

Given, the equation of line is 3x + 4y = m.

3x + 4y - m = 0

The equation of circle is : x² + y² = 10x

⇒ x² - 10x + y² = 0

Adding 25 to both sides

⇒ x² - 10x + 25 + y² = 25

⇒ (x - 5)² + y² = 5²                                  [(x-a)² = x² + y² -2ax]

Centre at (5,0).

So, distance of the line from the circle is given as:

Magnitude of (3×5 + 4×0 - m)/(√(4² + 3²))

= (15 - m)/5

This magnitude is equal to √25 = ±5

When 5 is negative,

So, (15 - m)/5 = -5

⇒ 15 - m = -25

⇒ m = 25+15 = 40

When 5 is positive,

So, (15 - m)/5 = 5

⇒ 15 - m = 25

⇒ m = 15-25 = -10

So, values are 40, -10.