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Parametric Equation of Tangent

If y1, y2 are...

Question

If $y_{1}, y_{2}$ are the ordinates of two points P and Q on the parabola and $y_{3}$ is the ordinate of the point of intersection of tangents at P and Q, then

A

are in A.P

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B

are in A.P

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C

are in G.P

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D

are in G.P

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Solution

The correct option is B are in A.P
Let $y_{1} = 2 a t_{1}, y_{2} = 2 a t_{2}, t h e n y_{3} = a (t_{1} + t_{2})$
$y_{3} = a (t_{1} + t_{2}) = \frac{2 a t_{1} + 2 a t_{2}}{2} = \frac{y_{1} + y_{2}}{2} . ∴ y_{1}, y_{3}, y_{2} a r e i n A . P$

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Similar questions

Q. If the tangents to the parabola

y^{2} = 4 a x

(x_{1}, y_{1}), (x_{2}, y_{2})

(x_{3}, y_{3}),

Q. If the points

(x_{1}, y_{1}), (x_{2}, y_{2})

(x_{3}, y_{3})

are collinear, show that

\sum (\frac{y_{1} - y_{2}}{x_{1} x_{2}}) = 0

\frac{y_{1} - y_{2}}{x_{1} x_{2}} + \frac{y_{2} - y_{3}}{x_{2} x_{3}} + \frac{y_{3} - y_{1}}{x_{3} x_{1}} = 0

Q. If three points

(x_{1}, y_{1}), (x_{2}, y_{2})

(x_{3}, y_{3})

lie on the same line, prove that

\frac{y_{2} - y_{3}}{x_{2} x_{3}} + \frac{y_{3} - y_{1}}{x_{3} x_{1}} + \frac{y_{1} - y_{2}}{x_{1} x_{2}} = 0

x_{1}, x_{2}, x_{3}

y_{1}, y_{2}, y_{3}

are in GP with same common ratio, then

(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})

x_{1}, x_{2}, x_{3}

y_{1}, y_{2}, y_{3}

are in G.P. with same common ratio, then the points

P (x_{1}, y_{1})

Q (x_{2}, y_{2})

R (x_{3}, y_{3})

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