 QUESTION

# Write an equation of a horizontal parabola with the given vertex and passing through the given point. Vertex at (4,3) and passing through (6,2).

Hint: Analyse the given. Substitute the coordinates of vertex in the general equation of Horizontal parabola, and you’ll get an equation. In that equation, substitute the coordinates of the point in which the vertex passes through and find out the value of a. Now, try to get the final answer.

Let the equation of the horizontal parabola is ${(y - h)^2} = 4a(x - k)$ because it is passing through (6,2) which is right side of the parabola

i.e, parabola is opening right side

So the vertex of the parabola is (k,h)

Compare the vertex from the given vertex i.e,(4,3)

$\Rightarrow k = 4,\;h = 3$

So, the equation of parabola is,

${(y - 3)}^2 = 4a(x - 4)$

Now it is passing through (6,2) which satisfy the parabola

$\Rightarrow {\left( {2 - 3} \right)^2} = 4a\left( {6 - 4} \right)$

$\Rightarrow 1 = 4a \times 2 \Rightarrow a = \frac{1}{8} \Rightarrow 4a = \frac{1}{2}$

$\Rightarrow$ Equation of parabola is,

${\left( {y - 3} \right)^2} = \frac{1}{2}\left( {x - 4} \right)$

So, this is your required equation.

NOTE: - In this type of problem first check from given condition parabola is opening which side, then satisfying the condition according to given problem statement you will get your answer.