Q. 125.0( 4 Votes )

# The line 2x – y + 4 = 0 cuts the parabola y^{2} = 8x in P and Q. The mid - point of PQ is

A. (1, 2)

B. (1, - 2)

C. (- 1, 2)

D. (- 1, - 2)

Answer :

Given that the line 2x - y + 4 = 0 cuts the parabola y^{2} = 8x at P, Q. We need to find the midpoint of PQ.

Substituting y = 2x + 4 in the equation of parabola.

⇒ (2x + 4)^{2} = 8x

⇒ 4x^{2} + 16x + 16 = 8x

⇒ 4x^{2} + 8x + 16 = 0

⇒ x^{2} + 2x + 4 = 0

Let x_{1}, x_{2} be the roots. Then, x_{1} + x_{2} = - 2

Now substituting in the equation of the line we get,

⇒

⇒

⇒ y^{2} - 4y + 16 = 0

Let y_{1}, y_{2} be the roots. Then, y_{1} + y_{2} = 4

Let us assume P be (x_{1}, y_{1}) and Q be (x_{2}, y_{2}) and R be the midpoint of PQ.

⇒

⇒

⇒ R = (- 1, 2)

∴The correct option is C

Rate this question :

The equation 16x^{2} + y^{2} + 8xy – 74x – 78y + 212 = 0 represents

In the parabola y^{2} = 4ax, the length of the chord passing through the vertex and inclined to the axis at is

Fill in the Blanks

The equation of the parabola having focus at (–1, –2) and the directrix x – 2y + 3 = 0 is ____________ .

Mathematics - ExemplarShow that the set of all points such that the difference of their distances from (4, 0)and (– 4, 0) is always equal to 2 represent a hyperbola.

Mathematics - ExemplarIf the line y = mx + 1 is tangent to the parabola y2 = 4x then find the value of m.

Mathematics - ExemplarFind the coordinates of a point on the parabola y2 = 8x whose focal distance is 4.

Mathematics - ExemplarThe line 2x – y + 4 = 0 cuts the parabola y^{2} = 8x in P and Q. The mid - point of PQ is

Find the equations of the lines joining the vertex of the parabola y^{2} = 6x to the point on it which have abscissa 24.

If the line y = mx + 1 is tangent to the parabola y^{2} = 4x, then find the value of m.