Here you will learn some parabola examples for better understanding of parabola concepts.
Example 1 : The length of latus rectum of a parabola, whose focus is (2, 3) and directrix is the line x โ 4y + 3 = 0 is โ
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 : \(\because\) ย ย Point (k-1, k) lies inside the parabola \(y^2\) = 4x.
\(\therefore\) ย ย \({y_1}^2 โ 4ax_1\) < 0
\(\implies\) ย ย \(k^2\) โ 4(k-1) < 0
\(\implies\) ย ย \(k^2\) โ 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)
\(\therefore\) ย ย 10 = 4m + \(9\over 4m\) \(\implies\) 16\(m^2\) โ 40m + 9 = 0
m = \(1\over 4\), \(9\over 4\)
\(\therefore\) ย ย Equation of tangentโs 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 โ 2a(x+h) = \(k^2\) โ 4ah.
Since it passes through (p,q)
\(\therefore\) ย ย qk โ 2a(p+h) = \(k^2\) โ 4ah
\(\therefore\) ย ย Required locus is \(y^2\) โ 2ax โ qy + 2ap = 0
Practice these given parabola examples to test your knowledge on concepts of parabola.
