Question

the number of common chords of parabola x=y2-6y+1 and y=x2-6x+1 are
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Answer 1

Given:

The equations of parabola x=y2-6y+1 and y=x2-6x+1

To find:

The number of common chords of parabola x=y2-6y+1 and y=x2-6x+1 are

​Solution:

From given, we have,

The equations of parabola x=y2-6y+1 and y=x2-6x+1

Let,

x = y² - 6y + 1 ......(1)

y = x² - 6x + 1 ......(2)

(1) - (2) gives,

x - y = (y² - 6y + 1) - (x² - 6x + 1)

x - y = y² - 6y + 1 - x² + 6x - 1

x - y = (y² - x²) - 6(y - x) - 1 + 1

x - y = (y² - x²) - 6(y - x)

x - y = (y - x) (y + x) - 6(y - x)

- (y - x) = (y - x) [y + x - 6]

(y - x) [y + x - 6] + (y - x) = 0

(y - x) [y + x - 6 + 1] = 0

(y - x) [y + x - 5] = 0

∴ (y - x) = 0  and  [y + x - 5] = 0

⇒ y = x and x + y = 5

Therefore, there are 2 common chords y = x and x + y = 5