Question

the area of triangle formed by tangents and the chord of contact from (3,4) to y^2=2x is

Answer 1

Answer:

7√10 sq. units

Step-by-step explanation:

Hi,

Let the chord of contact from P(3, 4) to the parabola S=y² - 2x = 0 be AB.

So, Equation of chord of contact from (3, 4) to parabola y² = 2x , AB will be

S₁ = 0

i.e., 4y - x - 3 = 0

or

x = 4y -3

Finding the point of intersection of line AB with parabola, we get

y² = 2(4y - 3)

=> y² - 8y + 6 = 0

y₁, y₂ are roots

=> (y₁-y₂)² = (y₁+y₂)² - 4y₁y₂

= 8² - 4*6

=40

x₁ = 4y₁-3

x₂ = 4y₂-3

So, length of AB = √(x₁-x₂)² + (y₁-y₂)²

=√17(y₁-y₂)²

=√680,

Also perpendicular distance from point P(3, 4) to the line AB, height of the triangle = |16-3-3|/√4²+1

= 7/√17.

Hence, Area of the triangle

= 1/2* base* height

=1/2*√680*7/√17

=7√10 sq. units

Hope, it helped !