Question

If 3x + y - 3 = 0 is a tangent at vertex of parabola and the lines x + y - 7 = 0 and 2x + y - 3 = 0 are other tangents to same parabola, then coordinates of its focus is​

Answer 1

Answer:

If 3x + y - 3 = 0 is a tangent at vertex of parabola and the lines x + y - 7 = 0 and 2x + y - 3 = 0 are other tangents to same parabola.

Answer 2

Step-by-step explanation:

Given: Tangent line is $3x+y-3=0$ at the vertex of parabola, and lines $x+y-7=0$ and $2x+y-3=0$ are other tangents to the same parabola.

To find:  Coordinates of its focus.

For calculation of the coordinates,

We have $x+y-7=0$ and $2x+y=3=0$

Thus,

⇒ $1+ \frac{dy}{dx} - 0 =0$

⇒ $\frac{dy}{dx} = -1$

Also,

⇒ $2$ × $1 + \frac{dy}{dx} - 0 = 0$

⇒$2+\frac{dy}{dx} = 0$

⇒ $\frac{dy}{dx} = -2$

Now, $-1$ and $-2$ will satisfy the equation of the tangent at the vertex of the parabola.

i. For calculation of y,

We put $x=-1$

⇒ $3x-1+y-3=0$

⇒ $y=-5$

ii. For calculation of x,

we put $x=-2$

⇒ $3(-2) +x = 3$

⇒$-6 +x = 3$

⇒$x=16$

Hence, the coordinates of the focus will be $(16,-5)$.