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Question
Answer the following:
The tangent at point P on the parabola y2 = 4ax meets the y-axis in Q. If S is the focus, show that SP subtends a right angle at Q
Solution
Let P(`"at"_1^2`, 2at1) be a point on the parabola and S(a, 0) be the focus of parabola y2 = 4ax
Since the tangent passing through point
P meet Y-axis at point Q,
equation of tangent at P(`"at"_1^2`, 2at1) is
yt1 = x + `"at"_1^2` ...(i)
∵ Point Q lie on tangent
∴ put x = 0 in equation (i)
yt1 = `"at"_1^2`
y = at1
∴ Co-ordinate of point Q(0, at1)
S = (a, 0), P(`"at"_1^2`, 2at1), Q(0, at1)
Slope of SQ = `(y_2 - y_1)/(x_2 - x_1)`
= `("at"_1 - 0)/(0 - "a")`
= `("at"_1)/(-"a")`
= – t1
Slope of PQ = `(y_2 - y_1)/(x_2 - x_1)`
= `(2"at"_1 - "at"_1)/("at"_1^2)`
= `"at"_1/"at"_1^2`
= `1/"t"_1`
∵ Slope of SQ × Slope of PQ
= `-"t"_1 xx 1/"t"_1`
= – 1
∴ SP subtends a right angle at Q.
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