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

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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 $\in$ $\phi$

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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

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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

Practice these given parabola examples to test your knowledge on concepts of parabola.