Question

Let a line y = mx (m > 0) intersect the parabola y² = x at a point P other than the origin. Let the tangent to it at P meet the x-axis at point Q. If area(ΔOPQ) = 4 sq. units, then m is equal to ______.

Answer 1

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Answer :-

⇝ Value of m is 1/2 .

Solution :-

⇝ We are the equation of Parabola as y² = x .

When we compare it with its standard form .

⇝ y² = 4ax

⇝ y² = 4 ( 1/4 ) x

⇝ y² = x

We got the value of a = 1/4

Equation of tangent

⇝ ty = x + at²

⇝ ty = x +( 1/4)t²

⇝ ty = x + t²/4

For finding the value of coordinates of Q , we'll put y = 0 .

⇝ t(0) = x + t²/4

⇝ 0 = x + t²/4

⇝ x = - ( t²/4 )

So, coordinates of Q( - t²/4 , 0 )

⇝ Area of ∆ POQ = 4 sq unit

⇝ Area = 1/2 ( b × h )

⇝ base = t²/ 4. ; height = t/2

⇝ 4 = 1/2 ( t²/4 × t/2 )

⇝ 4³ = t³

⇝ t = 4 .

So coordinates of P ( t²/4, t/2) =( 4, 2 )

Value of m →

⇝ y = mx

⇝2 = m ×4

⇝ m = 2/4

⇝ m = 1/2 .

Answer 2

Answer:

Value of m is 1/2 .

Solution :-

⇝ We are the equation of Parabola as y² = x .

When we compare it with its standard form .

⇝ y² = 4ax

⇝ y² = 4 ( 1/4 ) x

⇝ y² = x

We got the value of a = 1/4

Equation of tangent

⇝ ty = x + at²

⇝ ty = x +( 1/4)t²

⇝ ty = x + t²/4

For finding the value of coordinates of Q , we'll put y = 0 .

⇝ t(0) = x + t²/4

⇝ 0 = x + t²/4

⇝ x = - ( t²/4 )

So, coordinates of Q( - t²/4 , 0 )

⇝ Area of ∆ POQ = 4 sq unit

⇝ Area = 1/2 ( b × h )

⇝ base = t²/ 4. ; height = t/2

⇝ 4 = 1/2 ( t²/4 × t/2 )

⇝ 4³ = t³

⇝ t = 4 .

So coordinates of P ( t²/4, t/2) =( 4, 2 )

Value of m →

⇝ y = mx

⇝2 = m ×4

⇝ m = 2/4

⇝ m = 1/2 .