# Parabola Questions

## The focal distance of a point on the parabola $$y^2$$ = 12x is 4. Find the abscissa of this point.

Solution : The given parabola is of form $$y^2$$ = 4ax. On comparing, we have 4a = 12 i.e a = 3. We know that the focal distance of any point (x, y) on $$y^2$$ = 4ax is x + a. Let the given point on the parabola $$y^2$$ = 12 x be (x, y). …

## The sum of the slopes of the tangent of the parabola $$y^2$$=4ax drawn from the point (2,3) is

Solution : The equation of tangent to the parabola $$y^2$$ = 4ax is y = mx + $$a\over m$$. Since it is drawn from point (2,3) Therefore it lies on tangent y = mx + $$a\over m$$. $$\implies$$ 3 = 2m + $$a\over m$$ $$\implies$$ 3m = 2$$m^2$$ + a $$\implies$$  2$$m^2$$ – 3m + …

## The slope of the line touching both the parabolas $$y^2$$ = 4x and $$x^2$$ = -32 is

Solution : for parabola, $$y^2$$ = 4x Let y = mx + $$1\over m$$ is tangent line and it touches the parabola $$x^2$$ = -32. $$\therefore$$ $$x^2$$ = -32(mx + $$1\over m$$) $$\implies$$ $$x^2 + 32mx + {32\over m}$$ = 0 Now, D = 0 because it touches the curve. $$\therefore$$ $$(32m)^2 – 4.{32\over m}$$ …

## 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 Similar …

## 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 …

## 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 $$\in$$ $$\phi$$ Similar Questions The slope of the line touching both the parabolas $$y^2$$ = 4x and …

## 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 perpendicular from focus to the directrix = 2 x |$${2-4(3)+3}\over {\sqrt{1+16}}$$| = $$14\over \sqrt{17}$$ Similar Questions The slope of the line touching both the parabolas $$y^2$$ = 4x and $$x^2$$ = -32 is Find the locus of middle point of the chord of the parabola …

## What is the equation of common tangent to the parabola $$y^2$$ = 4ax and $$x^2$$ = 4ay ?

Solution : The equation of tangent in slope form to $$y^2$$ = 4ax is y = mx + $$a\over m$$ Now, if it is common to both parabola, it also lies on second parabola then $$x^2$$ = 4a(mx + $$a\over m$$) $$mx^2 – 4am^2 – 4a^2$$ = 0 has equal roots. then its discriminant is …

## What is the equation of tangent to the parabola having slope m?

Solution : The Equation of tangent to the parabola having slope ‘m’, is y = mx + $$a\over m$$ ,  (m $$\ne$$ 0) and point of contact  is ($$a\over m^2$$, $$2a\over m$$). Similar Questions The sum of the slopes of the tangent of the parabola $$y^2$$=4ax drawn from the point (2,3) is Find the locus …