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Equation of Tangent in Slope Form

If the tangen...

Question

If the tangents at P(t1) and Q(t2) in the parabola $y^{2} = 4 a x$ meet in T, then find the coordinates of point T.

A

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B

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C

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D

None of these

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Solution

The correct option is C

Let's take a calculate coordinates of T by calculating tangents

at P $&$ Q and their point of intersection is point T

Tangent at P

$y (2 a t_{1}) = 2 a (x + a t_{1}^{2})$

$y t_{1} = x + a t_{1}^{2}$ ...............(1)

Similarly, tangent at Q

$y t_{2} = x + a t_{2}^{2}$ ................(2)

Solving equation (1) $&$ (2)

We get, $x = a t_{1} t_{2}$

$y = a (t_{1} + t_{2})$

Coordinates of $T (a t_{1} t_{2}, a (t_{1} + t_{2})$

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Similar questions

Q. The point of concurrency of the altitudes drawn from the vertices

A (a t_{1} t_{2}, a (t_{1} + t_{2})), B (a t_{2} t_{3}, a (t_{2} + t_{3}))

C (a t_{3} t_{1}, a (t_{3} + t_{1}))

of the triangle

A B C

t_{1} \neq t_{2} \neq t_{3})

t_{1} + t_{2} + t_{3} = - t_{1} t_{2} t_{3}

, then the orthocentre of the triangle formed by the points

(a t_{1} t_{2}, a (t_{1} + t_{2}))

(a t_{2} t_{3}, a (t_{2} + t_{3}))

(a t_{3} t_{1}, a (t_{3} + t_{1}))

x - b + λ y = 0

cuts the parabola

y^{2} = 4 a x

P (t_{1})

Q (t_{2})

b \in [2 a, 4 a]

λ \in R

, then the value(s)

[t_{1} t_{2}]

[.]

repreasents the greatest integer function and

t_{1}, t_{2}

are parametic points)

Q. The vertices of a triangle are

[a t_{1} t_{2}, a (t_{1} + t_{2})], [a t_{2} t_{3}, a (t_{2} + t_{3})], [a t_{3} t_{1}, a (t_{3} + t_{1})]

. Find the coordinates of its orthocentre.

x - b + λ y = 0

cuts the parabola

y^{2} = 4 a x

P (t_{1})

Q (t_{2})

b \in [2 a, 4 a]

λ \in R

, then the value(s)

[t_{1} t_{2}]

[.]

repreasents the greatest integer function and

t_{1}, t_{2}

are parametic points)

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