Question

The focal distance of a point on the parabola $y^2=4x$ and above its axis, is 10 units. Its coordinates are
A). (9, 6)
B). (25, 10)
C). (25, -10)
D). None of these

Answer 1

Hint: Before attempting this question one must have prior knowledge of parabola, the equation which represents the parabola is $y^2=4ax$, using this information will help you to approach towards the solution of the problem

Complete step-by-step solution -

According to the given information we have a parabola $y^2=4x$ whose focal distance is 10
We know that the general equation of parabola is $y^2=4ax$
Taking the given equation of parabola i.e. $y^2=4x$ as equation 1
Comparing the general equation of parabola with the given equation of parabola we get
$4x=4ax$
$\Rightarrow$a = 1
So the coordinates of focus of parabola will be (1, 0)
We have focus distance = 10
So focal distance = 10 =$\sqrt{{\left(x-1\right)}^2+{\left(y-0\right)}^2}$
Squaring both sides we get
$\Rightarrow$$${\left(10\right)}^2={\left(\sqrt{{\left(x-1\right)}^2+{\left(y-0\right)}^2}\right)}^2$$
$\Rightarrow$$$100={\left(x-1\right)}^2+{\left(y-0\right)}^2$$
$\Rightarrow$$$100=x^2+1-2x+y^2$$
Substituting the value of $y^2$equation 1
We get $$100=x^2+1-2x+4x$$
$\Rightarrow$$$99=x^2+2x$$
$\Rightarrow$$$x^2+2x-99=0$$
By the method of splitting the middle term method
We get $$x^2+\left(11-9\right)x-99=0$$
$\Rightarrow$$$x^2+11x-9x-99=0$$
$\Rightarrow$$$x\left(x+11\right)-9\left(x+11\right)=0$$
So $x=-11,9$
Since x can’t be negative
Now substituting the value of x in equation 1
For x = 9
$y^2=4\left(9\right)$
$\Rightarrow$$y^2=36$
$\Rightarrow$$$y=\sqrt{36}$$
So $y=6,-6$
Therefore the coordinates are (9, 6)
Hence option A is the correct option.

Note: In the above solution we came across the terms parabola and focal distance which can be explained as a curve which consists of a set of all points that exist at equal distance from a fixed point (focus) this curve is named as a parabola. The distance from the vertex to focus which is measured along the symmetry of the axis is called focal distance.