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Chapter 2: Problem 22

Prove that an infinite number of triangles can be inscribed in either of the parabolas \(y^{2}=4 a x\) and \(x^{2}=4 b y\) whose sides touch the other.

Short Answer

Expert verified
An infinite number of triangles can be inscribed in either of the parabolas whose sides touch the other. This is because any line drawn tangent to one parabola will intersect the other parabola perpendicularly, and thus create a triangle; there are an infinite number of such tangent lines possible, so an infinite number of such triangles can be formed.

Step by step solution

01

Writing the equations of the parabolas

First, write down the given equations of the parabolas. They are \(y^{2}=4 a x\) and \(x^{2}=4 b y\). These are the standard forms for parabolas that open rightwards and upwards, respectively.
02

Understanding parabolic properties

Understand that a line tangent to a parabola always intersects the directrix at a right angle. For a parabola in the form \(y^{2}=4 a x\) or \(x^{2}=4 b y\), any line \(mx-c\) will be a tangent if \(c=\pm2am^{3/2}\). This will touch the parabola at point \( (am^{2}, 2am) \). Similar properties apply to the second parabola.
03

Using tangent properties to find triangles

We know a tangent line can touch the parabola at any point. There are infinite points on a parabola where a tangent can touch, which means there are infinite tangent lines. As the other parabola intersects these lines at right angles (because it intersects the directrix at right angles), these lines form the sides of right-angled triangles, with the common tangent as the hypotenuse. Therefore, there are infinite such triangles as there are infinite tangent lines possible.

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Most popular questions from this chapter

The point of intersection of the lines \(\frac{x}{a}+\frac{y}{b}=1\) and \(\frac{x}{b}+\frac{y}{a}=1\) lies on: (a) \(x-y=0\) (b) \((x+y)(a+b)=2 a b\) (c) \((\lfloor x+m y)(a+b)=(l+m) a b\) (d) \((b x-m y)(a+b)=(l-m) a b\)

Show that the locus of the centroids of equilateral triangles inscribed in the parabola \(y^{2}=4 a x\) is the parabola \(9 y^{2}-4 x a+32 a^{2}=0 .\)

The base \(B C\) of a triangle \(A B C\) contains the points \(P\left(p_{1}, q_{1}\right)\) and \(Q\left(p_{2}, q_{2}\right)\) and the equation of sides \(A B\) and \(A C\) are \(p_{1} x+q_{1} y=1\) and \(q_{2} x+p_{2} y=1\) respectively. Prove that the equations of \(A P\) and \(A Q\) are \(\left(p_{1} q_{2}+q_{1} p_{2}-1\right)\left(p_{1} x+q_{1} y-1\right)=\left(p_{1}^{2}+q_{1}^{2}-1\right)\left(q_{2} x+p_{2} y-1\right)\) and \(2\left(p_{2} q_{2}-1\right)\left(p_{1} x+q_{1} y-1\right)=\left(p_{1} p_{2}+q_{1} q_{2}-1\right)\left(q_{2} x+p_{2} y-1\right)\) respectively.

The area of the triangle having vertices \((-2,1),(2,1)\) and irrational) is : (a) 2 (b) 3 (c) 4 (d) none of these

If \(p, x_{1}, x_{2}, \ldots, x_{i}, \ldots\) and \(q, y_{1}, y_{2}, \ldots, y_{i-. .}\) form two infinite arithmetic sequences with common difference \(a\) and \(b\) respectively, then find the locus of the point \((a, \beta)\), where $$ \alpha=\frac{1}{n} \sum_{i=1}^{n} x_{i} \quad \text { and } \quad \beta=\frac{1}{n} \sum_{i=1}^{n} y_{i} $$
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