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Question
If the tangent drawn from the point (–6, 9) to the parabola y2 = kx are perpendicular to each other, find k
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Solution
Given equation of the parabola is y2 = kx.
Comparing this equation with y2 = 4ax, we get
4a = k
∴ a = `"k"/4`
Equation of tangent to the parabola y2 = 4ax having slope m is y = `"m"x + "a"/"m"`
Since the tangent passes through the point (–6, 9),
9 = `- 6"m" + "k"/(4"m")`
∴ 36m = –24m2 + k
∴ 24m2 + 36m – k = 0
The roots m1 and m2 of this quadratic equation are the slopes of the tangents.
∴ m1m2 = `(-"k")/24`
Since the tangents are perpendicular to each other,
m1m2 = – 1
∴ `(-"k")/24` = – 1
∴ k = 24
Alternate method:
We know that, tangents drawn from a point on directrix are perpendicular.
∴ (–6, 9) lies on the directrix x = – a.
∴ – 6 = – a
∴ a = 6
Since 4a = k,
k = 4(6) = 24
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