The circle x2 + y2 – 2x – 6y + 2 = 0 intersects the parabola y2

1.	The circle x2 + y2 – 2x – 6y + 2 = 0 intersects the parabola y2 = 8x orthogonally at the point P. The equation of the tangent to the parabola at P can be(a) 2x – y + 1 = 0(b) 2x + y – 2 = 0(c) x + y – 4 = 0(d) x – y – 4 = 0
Answer» Correct option (a) 2x –y+1 = 0 Explanation: Let y = mx + 2/m be tangent to y² = 8x. Since circle intersects the parabola orthogonally. So this tangent is the normal for the circle. Every normal of the circle passes through its centre. So centre (1, 3). 3 - m + 2/m² - 3m + 2 = 0 (m - 2)(m -1) = 0 m = 1, 2 y = x + 2 or y = 2x + 1

The circle x2 + y2 – 2x – 6y + 2 = 0 intersects the parabola y2 = 8x orthogonally at the point P. The equation of the tangent to the parabola at P can be(a) 2x – y + 1 = 0(b) 2x + y – 2 = 0(c) x + y – 4 = 0(d) x – y – 4 = 0

Answer»

Correct option (a) 2x –y+1 = 0

Explanation:

Let y = mx + 2/m be tangent to y² = 8x. Since circle intersects the parabola orthogonally. So this tangent is the normal for the circle. Every normal of the circle passes through its centre. So centre (1, 3).

3 - m + 2/m² - 3m + 2 = 0

(m - 2)(m -1) = 0

m = 1, 2

y = x + 2 or y = 2x + 1

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The circle x2 + y2 – 2x – 6y + 2 = 0 intersects the parabola y2 = 8x orthogonally at the point P. The equation of the tangent to the parabola at P can be(a) 2x – y + 1 = 0(b) 2x + y – 2 = 0(c) x + y – 4 = 0(d) x – y – 4 = 0

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