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Chapter 2: Problem 9

If \(\left(a^{2}, a-2\right)\) be a point interior to the region of the parabola \(y^{2}=2 x\) bounded by the chord joining the points \((2,2)\) and \((8,-4)\), then find the set of all possible real values of \(a\).

Short Answer

Expert verified
The set of all real values of \(a\) possible is \(\sqrt{4 - \sqrt{7}} \leq a \leq \sqrt{4 + \sqrt{7}}\)

Step by step solution

01

Determine the equation of the given chord

The chord passes through the points (2,2) and (8,-4). Let's use two-point form of the line, which is given by \(y- y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x- x_1)\). So, substituting \(x_1 = 2, y_1 = 2, x_2 = 8, y_2 = -4\), we get: \(y-2 = \frac{-4 - 2}{8 - 2} (x-2)\) which simplifies to \(y = -x + 6\). This is the equation of the chord.
02

Obtain the equation of the parabola

The given parabola equation is \(y^2 = 2x\). This can simply be rewritten as \(x = \frac{y^2}{2}\). Now, substitute the value of \(y = -x+6\) from step 1 into the rewritten parabolic equation. It becomes: \(\frac{(-x+6)^2}{2} = x\)
03

Solve to find the values of x and y

The equation from step 2 is a quadratic equation. We can solve this to get the values of x. After simplifying the expression, we have \(x^2 - 6x + 9 - 2x = 0\) which is \(x^2 - 8x + 9 = 0\). By applying the quadratic formula, we get the roots \(x = 4\pm \sqrt{7}\). So, these are the x-coordinates of (2,2) and (8,-4) on the parabola. We can substitute these x-values into the equation from Step 2 to get the corresponding y-values.
04

Determine the range of a

We know that the point \((a^2, a-2)\) is interior to the region bounded by the chord, so the x-coordinate \(a^2\) is greater than \(4 - \sqrt{7}\) and less than \(4 + \sqrt{7}\). So the possible values of a are in between \(\sqrt{4 - \sqrt{7}}\) and \(\sqrt{4 + \sqrt{7}}\). Note that a cannot be negative since \(a^2\) would then be shifted to left of the origin, which is not possible in the given question.

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Most popular questions from this chapter

Let a given line \(L_{1}\) intersect the \(x\) and \(y\)-axes at \(P\) and \(Q\) respectively. Let another line \(L_{2}\) perpendicular to \(L_{1}\), cut the \(x\) and \(y\)-axes at \(R\) and \(S\), respectively. Show that the locus of the point of intersection of the lines \(P S\) and \(Q R\) is a circle passing through the origin. (iIT 87,3 )

A ray of light is sent along the line \(x-2 y+5=0 .\) Upon reaching the line \(3 x+2 y+7=0\) the ray is reflected from it. Find the equation of the line containing reflected ray.

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Find the equation of circle, which touches the y.axis and the parabolas \(y^{2}=x\) and \(2 y=2 x^{2}-5 x+1\) only at the point when they touch each other and at 00 other point.

Find the locus of the point of intersection of tangents drawn at the end points of a variable chord of the parabola \(y^{2}=4 a x\), which subtends a constant angle \(\tan ^{-1}(b)\) at the vertex of the parabola.
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