ParabolaHard
Question
If the tangent at the point P (x1, y1) to the parabola y2 = 4ax meets the parabola y2 = 4a (x + b) at Q & R, then the mid point of QR is -
Options
A.(x1 + b, y1 + b)
B.(x1 - b, y1 - b)
C.(x1, y1)
D.(x1 + b, y1)
Solution
Equation of tangent to y2 = 4ax at P(x1, y1) is
yy1 = 2a(x + x1)
⇒ 2ax - yy1 + 2ax1 = 0 ... (i)
Let (h, k) be mid point of chord QR.
Then equation of QR is
ky - 2a(x + h) - 4ab = k2 - 4a(h + b)
⇒ - 2ax + ky + 2ah - k2 = 0 ... (ii)
Clearly (i) and (ii) represents same line.
y1 = k and 2ax1 = k2 - 2ah
2ax1 = y12 - 2ah
2ax1 = 4ax1 - 2ah ⇒ x1 = h
∴ mid point of QR is (x1, y1)
yy1 = 2a(x + x1)
⇒ 2ax - yy1 + 2ax1 = 0 ... (i)
Let (h, k) be mid point of chord QR.
Then equation of QR is
ky - 2a(x + h) - 4ab = k2 - 4a(h + b)
⇒ - 2ax + ky + 2ah - k2 = 0 ... (ii)
Clearly (i) and (ii) represents same line.
y1 = k and 2ax1 = k2 - 2ah
2ax1 = y12 - 2ah
2ax1 = 4ax1 - 2ah ⇒ x1 = h
∴ mid point of QR is (x1, y1)
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