# PARABOLA EXAMPLES

Example 1 : The length of latus rectum of a parabola, whose focus is (2, 3) and directrix is the line x - 4y + 3 = 0 is -

Solution : The length of latus rectum = 2 x perp. from focus to the dirctrix

= 2 x |$${2-4(3)+3}\over {\sqrt{1+16}}$$| = $$14\over \sqrt{17}$$

Example 2 : Find the value of k for which the point (k-1, k) lies inside the parabola $$y^2$$ = 4x.

Solution : $$\because$$     Point (k-1, k) lies inside the parabola $$y^2$$ = 4x.

$$\therefore$$     $${y_1}^2 - 4ax_1$$ < 0

$$\implies$$     $$k^2$$ - 4(k-1) < 0

$$\implies$$     $$k^2$$ - 4k + 4 < 0

$$(k-2)^2$$ < 0 $$\implies$$ k $$\in$$ $$\phi$$

Example 3 : Find the equation of the tangents to the parabola $$y^2$$ = 9x which go through the point (4,10).

Solution : Equation of tangent to the parabola $$y^2$$ = 9x is

y = mx + $$9\over 4m$$

Since it passes through (4,10)

$$\therefore$$     10 = 4m + $$9\over 4m$$ $$\implies$$ 16$$m^2$$ - 40m + 9 = 0

m = $$1\over 4$$, $$9\over 4$$

$$\therefore$$     Equation of tangent's are y = $$x\over 4$$ + 9 & y = $$9x\over 4$$ + 1

Example 4 : Find the locus of middle point of the chord of the parabola $$y^2$$ = 4ax which pass through a given (p,q).

Solution : Let P(h,k) be the mid point of chord of the parabola $$y^2$$ = 4ax,

so equation of chord is yk - 2a(x+h) = $$k^2$$ - 4ah.

Since it passes through (p,q)

$$\therefore$$     qk - 2a(p+h) = $$k^2$$ - 4ah

$$\therefore$$     Required locus is $$y^2$$ - 2ax - qy + 2ap = 0