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प्रश्न
Answer the following:
Show that the two tangents drawn to the parabola y2 = 24x from the point (−6, 9) are at the right angle
उत्तर
Given equation of the parabola is y2 = 24x
Comparing this equation with y2 = 4ax, we get
4a = 24
∴ a = `24/4` = 6
Equation of tangent to the parabola y2 = 4ax having slope m is y = `"m"x + "a"/"m"`.
∴ y = `"m"x + 6/"m"`
But, (– 6, 9) lies on the tangent
∴ 9 = `-6"m" + 6/"m"`
∴ 9m = – 6m2 + 6
∴ 6m2 + 9m – 6 = 0
The roots m1 and m2 of this quadratic equation are the slopes of the tangents.
∴ m1m2 = `(-6)/6` = – 1
∴ Tangents drawn to the parabola y2 = 24x from the point (– 6, 9) are at right angle.
Alternate method:
Comparing the given equation with y2 = 4ax, we get
4a = 24
∴ a = 6
Equation of the directrix is x = – 6.
The given point lies on the directrix.
Since tangents are drawn from a point on the directrix are perpendicular,
Tangents drawn to the parabola y2 = 24x from the point (– 6, 9) are at the right angle.
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