Question

Q no 7
7. The tangent at the point P(x1, y1) to the parabola y2 = 4ax meets the parabola y2 = 4a (x + b) at Q and R, the coordinates of the mid-point of QR are
(A) (x1 - a, y1 + b) (B) (x1, y1)
(C) (x1 + b, y1 + a) (D) (x1 - b, y1 - b)

Answer 1

Answer:

B

Step-by-step explanation:

Hi,

Given that the tangent at the point P(x₁,y₁) to the parabola y²=4ax

meets the parabola y²=4a(x+b), at Q and R.

Equation of the tangent at the point P(x₁,y₁) to the parabola y²=4ax

is given by

              yy₁ = 2a(x+x₁)

But the above line intersects the parabola y²=4a(x+b), at Q(x₂, y₂) and

R(x₃, y₃).

Now finding the point of intersection of the tangent line with the parabola,

we get

[2a(x+x₁)/y₁]² = 4a(x+b)

On simplifying, we get

ax² + x(2ax₁-y₁²) + ax₁²-y₁²b = 0

For the above equation, roots are x₂ and x₃.

=> x₂ + x₃ = (y₁² - 2ax₁)/a.

But P(x₁, y₁) lies on the parabola y² = 4ax,

hence y₁² = 4ax₁

=>x₂ + x₃ =(4ax₁ - 2ax₁)/a = 2ax₁/a = 2x₁

=> (x₂ + x₃)/2 = x₁

Substituting x₁, we get y-coordinate of the midpoint as y₁.

Hence , the coordinates of the midpoint of the chord QR are P(x₁, y1).

Hope, it helped !

Answer 2

Answer:

Step-by-step explanation:

Hope it helps